Context modeling of side information for reduced secondary transforms in video

ABSTRACT

A video processing method is described. The method includes performing a conversion between a video region of a video and a coded representation of the video, wherein the performing of the conversion includes configuring, based on a partition type of the video region, a context model for coding a first bin, the first bin and a second bin included in a bin string corresponding to an index of a secondary transform tool, wherein the index indicates an applicability of the secondary transform tool and/or a kernel information of the secondary transform tool, and wherein the secondary transform tool includes applying, during encoding, a forward secondary transform to an output of a forward primary transform applied to a residual of a video block prior to quantization, or applying, during decoding, an inverse secondary transform to an output of dequantization to the video block before applying an inverse primary transform.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of International Patent ApplicationNo. PCT/CN2020/109476, filed on Aug. 17, 2020, which claims the priorityto and benefits of International Patent Application Nos.PCT/CN2019/101230 filed on Aug. 17, 2019, PCT/CN2019/107904 filed onSep. 25, 2019 and PCT/CN2019/127829 filed on Dec. 24, 2019. All theaforementioned patent applications are hereby incorporated by referencein their entireties.

TECHNICAL FIELD

This patent document relates to video processing techniques, devices andsystems.

BACKGROUND

In spite of the advances in video compression, digital video stillaccounts for the largest bandwidth use on the internet and other digitalcommunication networks. As the number of connected user devices capableof receiving and displaying video increases, it is expected that thebandwidth demand for digital video usage will continue to grow.

SUMMARY

Devices, systems and methods related to digital video processing, andspecifically, to context modeling for residual coding in video coding.The described methods may be applied to both the existing video codingstandards (e.g., High Efficiency Video Coding (HEVC)) and future videocoding standards or video codecs.

In one representative aspect, the disclosed technology may be used toprovide a method for video processing. This method includes performing aconversion between a video region of a video and a coded representationof the video, wherein the performing of the conversion includesconfiguring, based on a partition type of the video region, a contextmodel for coding a first bin, the first bin and a second bin included ina bin string corresponding to an index of a secondary transform toolapplied to the video region, wherein the index indicates anapplicability of the secondary transform tool and/or a kernelinformation of the secondary transform tool, and wherein the secondarytransform tool includes applying, during encoding, a forward secondarytransform to an output of a forward primary transform applied to aresidual of a video block prior to quantization, or wherein thesecondary transform tool includes applying, during decoding, an inversesecondary transform to an output of dequantization to the video blockbefore applying an inverse primary transform.

In another representative aspect, the disclosed technology may be usedto provide a method for video processing. This method includesdetermining, for a current video block of a video comprising sub-blocks,based on a coding condition of the current video block, a zero-outregion in which coefficients are zeroed out; and performing a conversionbetween the current video block and a coded representation of the videobased on the determining, wherein the conversion includes applying asecondary transform tool, and wherein the secondary transform toolincludes applying, during encoding, a forward secondary transform to anoutput of a forward primary transform applied to a residual of a videoblock prior to quantization, or wherein the secondary transform toolincludes applying, during decoding, an inverse secondary transform to anoutput of dequantization to the video block before applying an inverseprimary transform.

In yet another representative aspect, the disclosed technology may beused to provide a method for video processing. This method includesdetermining, for a conversion between a current video block having adimension N×N of a video and a coded representation of the video, to usea transform matrix and/or an inverse transform matrix with a reducedsize that is smaller than N×N in an application of a secondary transformtool to the current video block of a video; and performing a conversionbetween the video and a coded representation of the video, wherein thesecondary transform tool includes applying, during encoding, a forwardsecondary transform to an output of a forward primary transform appliedto a residual of a video block prior to quantization, or wherein thesecondary transform tool includes applying, during decoding, an inversesecondary transform to an output of dequantization to the video blockbefore applying an inverse primary transform.

In yet another representative aspect, the disclosed technology may beused to provide a method for video processing. This method includesdetermining, for a conversion between a current video block of a videoand a coded representation of the video, based on a rule, to use aresidual pattern among one or more residual patterns, wherein each ofthe one or more residual patterns corresponds to a mask that providesinformation about positions of zeroed out samples; and performing aconversion between the video and a coded representation of the videobased on the determining.

In yet another representative aspect, the disclosed technology may beused to provide a method for video processing. This method includesdetermining, based on a type of a primary transform, a secondarytransform tool associated with a current video block of a video; andperforming a conversion between the video and a coded representation ofthe video, wherein the secondary transform tool includes applying,during encoding, a forward secondary transform to an output of a forwardprimary transform applied to a residual of a video block prior toquantization, or wherein the secondary transform tool includes applying,during decoding, an inverse secondary transform to an output ofdequantization to the video block before applying an inverse primarytransform.

In yet another representative aspect, the disclosed technology may beused to provide a method for video processing. This method includesdetermining to use multiple sets of transform matrices to be used toprocess a video region of a video; and performing a conversion betweenthe video and a coded representation of the video, and wherein the codedrepresentation includes an indication of which set among the multiplesets is used for the video region.

In yet another example aspect, a method of video processing isdisclosed. The method includes determining, based on a rule, a transformmatrix or an inverse transform matrix to be used in an application of asecondary transform tool to a current video block of a video; andperforming a conversion between the video and a coded representation ofthe video, wherein the secondary transform tool includes applying,during encoding, a forward secondary transform to an output of a forwardprimary transform applied to a residual of a video block prior toquantization, or wherein the secondary transform tool includes applying,during decoding, an inverse secondary transform to an output ofdequantization to the video block before applying an inverse primarytransform, wherein the rule specifies to determine coefficients of thetransform matrix or the inverse transform matrix based on a variablespecifying a transform output length.

In yet another example aspect, a method of video processing isdisclosed. The method includes determining a transform matrix or aninverse transform matrix to be used in an application of a secondarytransform tool to a current video block of a video based on a rule; andperforming a conversion between the video and a coded representation ofthe video, wherein the secondary transform tool includes applying,during encoding, a forward secondary transform to an output of a forwardprimary transform applied to a residual of a video block prior toquantization, or wherein the secondary transform tool includes applying,during decoding, an inverse secondary transform to an output ofdequantization to the video block before applying an inverse primarytransform, wherein the rule specifies to determine the transform matrixor the inverse transform matrix by using a retraining process.

In yet another representative aspect, the above-described method isembodied in the form of processor-executable code and stored in acomputer-readable program medium.

In yet another representative aspect, a device that is configured oroperable to perform the above-described method is disclosed. The devicemay include a processor that is programmed to implement this method.

In yet another representative aspect, a video decoder apparatus mayimplement a method as described herein.

The above and other aspects and features of the disclosed technology aredescribed in greater detail in the drawings, the description and theclaims.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a block diagram of an example encoder.

FIG. 2 shows an example of 67 intra prediction modes.

FIG. 3 shows an example of ALWIP for 4×4 blocks.

FIG. 4 shows an example of ALWIP for 8×8 blocks.

FIG. 5 shows an example of ALWIP for 8×4 blocks.

FIG. 6 shows an example of ALWIP for 16×16 blocks.

FIG. 7 shows an example of four reference lines neighboring a predictionblock.

FIG. 8 shows an example of divisions of 4×8 and 8×4 blocks.

FIG. 9 shows an example of divisions all blocks except 4×8, 8×4 and 4×4.

FIG. 10 shows an example of a secondary transform in JEM.

FIG. 11 shows an example of the proposed reduced secondary transform(RST).

FIG. 12 shows examples of the forward and inverse reduced transforms.

FIG. 13 shows an example of a forward RST 8×8 process with a 16×48matrix.

FIG. 14 shows an example of a zero-out region for an 8×8 matrix.

FIG. 15 shows an example of sub-block transform modes SBT-V and SBT-H.

FIG. 16 shows an example of a diagonal up-right scan order for a 4×4coding group.

FIG. 17 shows an example of a diagonal up-right scan order for an 8×8block with coding groups of size 4×4.

FIG. 18 shows an example of a template used to select probabilitymodels.

FIG. 19 shows an example of two scalar quantizers used for dependentquantization.

FIG. 20 shows an example of a state transition and quantizer selectionfor the proposed dependent quantization process.

FIG. 21 an example of an 8×8 block with 4 coding groups.

FIGS. 22A to 22F show flowcharts of example methods for videoprocessing.

FIGS. 23A and 23B are block diagrams of examples of a hardware platformfor implementing a visual media decoding or a visual media encodingtechnique described in the present document.

FIG. 24A shows an example of one chroma blocks and FIG. 24B shows itscorresponding luma block under dual tree partition.

FIG. 25 shows an example of ‘CR’ position for DM derivation from thecorresponding luma block.

FIG. 26 shows an example of partial region allowed to have coefficients.

FIGS. 27A to 27H show flowcharts of example methods for videoprocessing.

DETAILED DESCRIPTION

Embodiments of the disclosed technology may be applied to existing videocoding standards (e.g., HEVC, H.265) and future standards to improvecompression performance. Section headings are used in the presentdocument to improve readability of the description and do not in any waylimit the discussion or the embodiments (and/or implementations) to therespective sections only.

2 Video Coding Introduction

Due to the increasing demand of higher resolution video, video codingmethods and techniques are ubiquitous in modern technology. Video codecstypically include an electronic circuit or software that compresses ordecompresses digital video, and are continually being improved toprovide higher coding efficiency. A video codec converts uncompressedvideo to a compressed format or vice versa. There are complexrelationships between the video quality, the amount of data used torepresent the video (determined by the bit rate), the complexity of theencoding and decoding algorithms, sensitivity to data losses and errors,ease of editing, random access, and end-to-end delay (latency). Thecompressed format usually conforms to a standard video compressionspecification, e.g., the High Efficiency Video Coding (HEVC) standard(also known as H.265 or MPEG-H Part 2), the Versatile Video Codingstandard to be finalized, or other current and/or future video codingstandards.

Video coding standards have evolved primarily through the development ofthe well-known ITU-T and ISO/IEC standards. The ITU-T produced H.261 andH.263, ISO/IEC produced MPEG-1 and MPEG-4 Visual, and the twoorganizations jointly produced the H.262/MPEG-2 Video and H.264/MPEG-4Advanced Video Coding (AVC) and H.265/HEVC standards. Since H.262, thevideo coding standards are based on the hybrid video coding structurewherein temporal prediction plus transform coding are utilized. Toexplore the future video coding technologies beyond HEVC, Joint VideoExploration Team (JVET) was founded by VCEG and MPEG jointly in 2015.Since then, many new methods have been adopted by JVET and put into thereference software named Joint Exploration Model (JEM) [3] [4]. In April2018, the Joint Video Expert Team (JVET) between VCEG (Q6/16) andISO/IEC JTC1 SC29/WG11 (MPEG) was created to work on the VVC standardtargeting at 50% bitrate reduction compared to HEVC.

2.1 Coding Flow of a Typical Video Codec

FIG. 1 shows an example of encoder block diagram of VVC, which containsthree in-loop filtering blocks: deblocking filter (DF), sample adaptiveoffset (SAO) and ALF. Unlike DF, which uses predefined filters, SAO andALF utilize the original samples of the current picture to reduce themean square errors between the original samples and the reconstructedsamples by adding an offset and by applying a finite impulse response(FIR) filter, respectively, with coded side information signaling theoffsets and filter coefficients. ALF is located at the last processingstage of each picture and can be regarded as a tool trying to catch andfix artifacts created by the previous stages.

2.2 Intra Coding in VVC

2.2.1 Intra Mode Coding with 67 Intra Prediction Modes

To capture the arbitrary edge directions presented in natural video, thenumber of directional intra modes is extended from 33, as used in HEVC,to 65. The additional directional modes are depicted as dotted arrows inFIG. 2 , and the planar and DC modes remain the same. These denserdirectional intra prediction modes apply for all block sizes and forboth luma and chroma intra predictions.

Conventional angular intra prediction directions are defined from 45degrees to −135 degrees in clockwise direction as shown in FIG. 2 . InVTM2, several conventional angular intra prediction modes are adaptivelyreplaced with wide-angle intra prediction modes for the non-squareblocks. The replaced modes are signaled using the original method andremapped to the indexes of wide angular modes after parsing. The totalnumber of intra prediction modes is unchanged, i.e., 67, and the intramode coding is unchanged.

In the HEVC, every intra-coded block has a square shape and the lengthof each of its side is a power of 2. Thus, no division operations arerequired to generate an intra-predictor using DC mode. In VVV2, blockscan have a rectangular shape that necessitates the use of a divisionoperation per block in the general case. To avoid division operationsfor DC prediction, only the longer side is used to compute the averagefor non-square blocks.

In addition to the 67 intra prediction modes, wide-angle intraprediction for non-square blocks (WAIP) and position dependent intraprediction combination (PDPC) methods are further enabled for certainblocks. PDPC is applied to the following intra modes without signalling:planar, DC, horizontal, vertical, bottom-left angular mode and its eightadjacent angular modes, and top-right angular mode and its eightadjacent angular modes.

2.2.2 Affine Linear Weighted Intra Prediction (ALWIP or Matrix-BasedIntra Prediction)

Affine linear weighted intra prediction (ALWIP, a.k.a. Matrix basedintra prediction (MIP)) is proposed in JVET-N0217.

2.2.2.1 Generation of the Reduced Prediction Signal by Matrix VectorMultiplication

The neighboring reference samples are firstly down-sampled via averagingto generate the reduced reference signal bdry_(red). Then, the reducedprediction signal pred_(red) is computed by calculating a matrix vectorproduct and adding an offset:

pred_(red) = A ⋅ bdry_(red) + b

Here, A is a matrix that has W_(red)·H_(red) rows and 4 columns if W=H=4and 8 columns in all other cases. b is a vector of size W_(red)·H_(red).

2.2.2.2 Illustration of the Entire ALWIP Process

The entire process of averaging, matrix vector multiplication and linearinterpolation is illustrated for different shapes in FIGS. 3-6 . Note,that the remaining shapes are treated as in one of the depicted cases.

1. Given a 4×4 block, ALWIP takes two averages along each axis of theboundary. The resulting four input samples enter the matrix vectormultiplication. The matrices are taken from the set S₀. After adding anoffset, this yields the 16 final prediction samples. Linearinterpolation is not necessary for generating the prediction signal.Thus, a total of (4·16)/(4·4)=4 multiplications per sample areperformed.

2. Given an 8×8 block, ALWIP takes four averages along each axis of theboundary. The resulting eight input samples enter the matrix vectormultiplication. The matrices are taken from the set S₁. This yields 16samples on the odd positions of the prediction block. Thus, a total of(8·16)/(8·8)=2 multiplications per sample are performed. After adding anoffset, these samples are interpolated vertically by using the reducedtop boundary. Horizontal interpolation follows by using the originalleft boundary.

3. Given an 8×4 block, ALWIP takes four averages along the horizontalaxis of the boundary and the four original boundary values on the leftboundary. The resulting eight input samples enter the matrix vectormultiplication. The matrices are taken from the set S₁. This yields 16samples on the odd horizontal and each vertical positions of theprediction block. Thus, a total of (8·16)/(8·4)=4 multiplications persample are performed. After adding an offset, these samples areinterpolated horizontally by using the original left boundary.

4. Given a 16×16 block, ALWIP takes four averages along each axis of theboundary. The resulting eight input samples enter the matrix vectormultiplication. The matrices are taken from the set S₂. This yields 64samples on the odd positions of the prediction block. Thus, a total of(8·64)/(16·16)=2 multiplications per sample are performed. After addingan offset, these samples are interpolated vertically by using eightaverages of the top boundary. Horizontal interpolation follows by usingthe original left boundary. The interpolation process, in this case,does not add any multiplications. Therefore, totally, twomultiplications per sample are required to calculate ALWIP prediction.

For larger shapes, the procedure is essentially the same and it is easyto check that the number of multiplications per sample is less thanfour.

For W×8 blocks with W>8, only horizontal interpolation is necessary asthe samples are given at the odd horizontal and each vertical positions.

Finally for W×4 blocks with W>8, let A_kbe the matrix that arises byleaving out every row that corresponds to an odd entry along thehorizontal axis of the downsampled block. Thus, the output size is 32and again, only horizontal interpolation remains to be performed.

The transposed cases are treated accordingly.

When constructing the MPM list for an intra predicted block, if aneighboring block is coded in MIP mode, the MIP mode will be mapped tothe intra prediction mode by using the following table. Here, assumingthe width and height of the neighboring luma block are widthNeig andheightNeig respectively, MipSizeId is derived as follows:

-   -   If both widthNeig and heightNeig are equal to 4, MipSizeId is        set equal to 0.    -   Otherwise, if both widthNeig and heightNeig are less than or        equal to 8, MipSizeId is set equal to 1.    -   Otherwise, MipSizeId is set equal to 2.

TABLE 8-4 Sepcification of mapping between MIP and intra predictionmodes MipSizeId IntraPredModeY[ xNbX ][ yNbX ] 0 1 2 0 0 0 1 1 18 1 1 218 0 1 3 0 1 1 4 18 0 18 5 0 22 0 6 12 18 1 7 0 18 0 8 18 1 1 9 2 0 5010 18 1 0 11 12 0 12 18 1 13 18 0 14 1 44 15 18 0 16 18 50 17 0 1 18 0 019 50 20 0 21 50 22 0 23 56 24 0 25 50 26 66 27 50 28 56 29 50 30 50 311 32 50 33 50 34 502.2.2.3 Syntax and Semantics7.3.6.5 Coding Unit Syntax

Descriptor coding_unit( x0, y0, cbWidth, cbHeight, treeType ) {  if(tile_group_type != I | | sps_ibc_enabled_flag ) {   if( treeType !=DUAL_TREE_CHROMA )    cu_skip_flag[ x0 ][ y0 ] ae(v)   if( cu_skip_flag[x0 ][ y0 ] = = 0 && tile_group_type != I )    pred_mode_flag ae(v)   if(( ( tile_group_type = = I && cu_skip_flag[ x0 ][ y0 ] = =0 ) | |    (tile_group_type != I && CuPredMode[ x0 ][ y0 ] != MODE_INTRA ) ) &&   sps_ibc_enabled_flag )    pred_mode_ibc_flag ae(v)  }  if(CuPredMode[ x0 ][ y0 ] = = MODE_INTRA ) {   if( sps_pcm_enabled_flag &&   cbWidth >= MinIpcmCbSizeY && cbWidth <= MaxIpcmCbSizeY &&   cbHeight >= MinIpcmCbSizeY && cbHeight <= MaxIpcmCbSizeY )   pcm_flag[ x0 ][ y0 ] ae(v)   if( pcm_flag[ x0 ][ y0 ] ) {    while(!byte_aligned( ) )     pcm_alignment_zero_bit f(1)    pcm_sample(cbWidth, cbHeight, treeType)   } else {    if( treeType = = SINGLE_TREE| | treeType = = DUAL_TREE_LUMA ) {     if( abs( Log2( cbWidth ) − Log2(cbHeight ) ) <= 2 )      intra_lwip_flag[ x0 ][ y0 ] ae(v)     if(intra_lwip_flag[ x0 ][ y0 ] ) {       intra_lwip_mpm_flag[ x0 ][ y0 ]ae(v)      if( intra_lwip_mpm_flag[ x0 ][ y0 ] )      intra_lwip_mpm_idx[ x0 ][ y0 ] ae(v)      else      intra_lwip_mpm_remainder[ x0 ][ y0 ] ae(v)     } else {      if( (y0 % CtbSizeY ) > 0 )       intra_luma_ref_idx[ x0 ][ y0 ] ae(v)      if(intra_luma_ref_idx[ x0 ][ y0 ] = = 0 &&       ( cbWidth <= MaxTbSizeY || cbHeight <= MaxTbSizeY ) &&       ( cbWidth * cbHeight > MinTbSizeY *MinTbSizeY ))       intra_subpartitions_mode_flag[ x0 ][ y0 ] ae(v)     if( intra_subpartitions_mode_flag[ x0 ][ y0 ] = = 1 &&      cbWidth <= MaxTbSizeY && cbHeight <= MaxTbSizeY )      intra_subpartitions_split_flag[ x0 ][ y0 ] ae(v)      if(intra_luma_ref_idx[ x0 ][ y0 ] = = 0 &&      intra_subpartitions_mode_flag[ x0 ][ y0 ] = = 0 )      intra_luma_mpm_flag[ x0 ][ y0 ] ae(v)      if(intra_luma_mpm_flag[ x0 ][ y0 ] )       intra_luma_mpm_idx[ x0 ][ y0 ]ae(v)      else       intra_luma_mpm_remainder[ x0 ][ y0 ] ae(v)     }   }    if( treeType = = SINGLE_TREE | | treeType = = DUAL_TREE_CHROMA )    intra_chroma_pred_mode[ x0 ][ y0 ] ae(v)   }  } else if( treeType !=DUAL_TREE_CHROMA ) { /* MODE_INTER or MODE_IBC */ ...  } }2.2.3 Multiple Reference Line (MRL)

Multiple reference line (MRL) intra prediction uses more reference linesfor intra prediction. In FIG. 7 , an example of 4 reference lines isdepicted, where the samples of segments A and F are not fetched fromreconstructed neighbouring samples but padded with the closest samplesfrom Segment B and E, respectively. HEVC intra-picture prediction usesthe nearest reference line (i.e., reference line 0). In MRL, 2additional lines (reference line 1 and reference line 3) are used.

The index of selected reference line (mrl_idx) is signaled and used togenerate intra predictor. For reference line index, which is greaterthan 0, only include additional reference line modes in MPM list andonly signal MPM index without remaining mode. The reference line indexis signaled before intra prediction modes, and Planar and DC modes areexcluded from intra prediction modes in case a nonzero reference lineindex is signaled.

MRL is disabled for the first line of blocks inside a CTU to preventusing extended reference samples outside the current CTU line. Also,PDPC is disabled when additional line is used.

2.2.4 Intra Sub-Block Partitioning (ISP)

In JVET-M0102, ISP is proposed, which divides luma intra-predictedblocks vertically or horizontally into 2 or 4 sub-partitions dependingon the block size dimensions, as shown in Table 1. FIG. 8 and FIG. 9show examples of the two possibilities. All sub-partitions fulfill thecondition of having at least 16 samples. For block sizes, 4×N or N×4(with N>8), if allowed, the 1×N or N×1 sub-partition may exist.

TABLE 1 Number of sub-partitions depending on the block size (denotedmaximum transform size by maxTB Size) Splitting Number of directionBlock Size Sub-Partitions N/A minimum transform size Not divided 4 × 8:4 × 8 and 8 × 4 2 horizontal 8 × 4: vertical Signaled If neither 4 × 8nor 8 × 4, and W <= 4 maxTBSize and H <= maxTBSize Horizontal If notabove cases and H > maxTBSize 4 Vertical If not above cases and H >maxTBSize 4

For each of these sub-partitions, a residual signal is generated byentropy decoding the coefficients sent by the encoder and then invertquantizing and invert transforming them. Then, the sub-partition isintra predicted and finally the corresponding reconstructed samples areobtained by adding the residual signal to the prediction signal.Therefore, the reconstructed values of each sub-partition will beavailable to generate the prediction of the next one, which will repeatthe process and so on. All sub-partitions share the same intra mode.

TABLE 2 Specification of trTypeHor and trTypeVer depending onpredModeIntra predModeIntra trTypeHor trTypeVer INTRA_PLANAR, ( nTbW >=4 && ( nTbH >= 4 && INTRA_ANGULAR31, nTbW <= 16 ) ? nTbH <= 16 ) ?INTRA_ANGULAR32, DST-VII : DCT-II DST-VII : DCT-II INTRA_ANGULAR34,INTRA_ANGULAR36, INTRA_ANGULAR37 INTRA_ANGULAR33, DCT-II DCT-IIINTRA_ANGULAR35 INTRA_ANGULAR2, ( nTbW >= 4 && DCT-II INTRA_ANGULAR4, .. . , nTbW <= 16 ) ? INTRA_ANGULAR28, DST-VII : DCT-II INTRA_ANGULAR30,INTRA_ANGULAR39, INTRA_ANGULAR41, . . . , INTRA_ANGULAR63,INTRA_ANGULAR65 INTRA_ANGULAR3, DCT-II ( nTbH >= 4 && INTRA_ANGULAR5, .. . , nTbH <= 16 ) ? INTRA_ANGULAR27, DST-VII : DCT-II INTRA_ANGULAR29,INTRA_ANGULAR38, INTRA_ANGULAR40, . . . , INTRA_ANGULAR64,INTRA_ANGULAR662.2.4.1 Syntax and Semantics7.3.7.5 Coding Unit Syntax

Descriptor coding_unit( x0, y0, cbWidth, cbHeight, treeType ) {  if(slice_type != I | | sps_ibc_enabled_flag ) {   if( treeType !=DUAL_TREE_CHROMA )    cu_skip_flag[ x0 ][ y0 ] ae(v)   if( cu_skip_flag[x0 ][ y0 ] = = 0 && slice_type != I )    pred_mode_flag ae(v)   if( ( (slice_type = = I && cu_skip_flag[ x0 ][ y0 ] = =0 ) | |    ( slice_type!= I && CuPredMode[ x0 ][ y0 ] != MODE_INTRA ) ) &&   sps_ibc_enabled_flag )    pred_mode_ibc_flag ae(v)  }  if(CuPredMode[ x0 ][ y0 ] = = MODE_INTRA ) {   if( sps_pcm_enabled_flag &&   cbWidth >= MinIpcmCbSizeY && cbWidth <= MaxIpcmCbSizeY &&   cbHeight >= MinIpcmCbSizeY && cbHeight <= MaxIpcmCbSizeY )   pcm_flag[ x0 ][ y0 ] ae(v)   if( pcm_flag[ x0 ][ y0 ] ) {    while(!byte_aligned( ) )     pcm_alignment_zero_bit f(1)    pcm_sample(cbWidth, cbHeight, treeType)   } else {    if( treeType = = SINGLE_TREE| | treeType = = DUAL_TREE_LUMA ) {     if( ( y0 % CtbSizeY ) > 0 )     intra_luma_ref_idx[ x0 ][ y0 ] ae(v)     if (intra_luma_ref_idx[ x0][ y0 ] = = 0 &&      ( cbWidth <= MaxTbSizeY | | cbHeight <= MaxTbSizeY) &&      ( cbWidth * cbHeight > MinTbSizeY * MinTbSizeY ))     intra_subpartitions_mode_flag[ x0 ][ y0 ] ae(v)     if(intra_subpartitions_mode_flag[ x0 ][ y0 ] = = 1 &&      cbWidth <=MaxTbSizeY && cbHeight <= MaxTbSizeY )     intra_subpartitions_split_flag[ x0 ][ y0 ] ae(v)     if(intra_luma_ref_idx[ x0 ][ y0 ] = = 0 &&     intra_subpartitions_mode_flag[ x0 ][ y0 ] = = 0 )     intra_luma_mpm_flag[ x0 ][ y0 ] ae(v)     if( intra_luma_mpm_flag[x0 ][ y0 ] )      intra_luma_mpm_idx[ x0 ][ y0 ] ae(v)     else     intra_luma_mpm_remainder[ x0 ][ y0 ] ae(v)    }    if( treeType = =SINGLE_TREE | | treeType = = DUAL_TREE_CHROMA )    intra_chroma_pred_mode[ x0 ][ y0 ] ae(v)   }  } else if( treeType !=DUAL_TREE_CHROMA ) { /* MODE_INTER or MODE_IBC */ ...  } ... }intra_subpartitions_mode_flag[x0] [y0] equal to 1 specifies that thecurrent intra coding unit is partitioned into NumIntraSubPartitions[x0][y0] rectangular transform block subpartitions.intra_subpartitions_mode_flag[x0] [y0] equal to 0 specifies that thecurrent intra coding unit is not partitioned into rectangular transformblock subpartitions.When intra_subpartitions_mode_flag[x0] [y0] is not present, it isinferred to be equal to 0.intra_subpartitions_split_flag[x0] [y0] specifies whether the intrasubpartitions split type is horizontal or vertical.When intra_subpartitions_split_flag[x0] [y0] is not present, it isinferred as follows:

-   -   If cbHeight is greater than MaxTbSizeY,        intra_subpartitions_split_flag[x0] [y0] is inferred to be equal        to 0.    -   Otherwise (cbWidth is greater than MaxTbSizeY),        intra_subpartitions_split_flag[x0] [y0] is inferred to be equal        to 1.        The variable IntraSubPartitionsSplitType specifies the type of        split used for the current luma coding block as illustrated in        Table 7-9. IntraSubPartitionsSplitType is derived as follows:    -   If intra_subpartitions_modeflag[x0] [y0] is equal to 0,        IntraSubPartitionsSplitType is set equal to 0.    -   Otherwise, the IntraSubPartitionsSplitType is set equal to        1+intra_subpartitions_split_flag[x0] [y0].

TABLE 7-9 Name association to IntraSubPartitionsSplitTypeIntraSubPartitionsSplitType Name of IntraSubPartitionsSplitType 0ISP_NO_SPLIT 1 ISP_HOR_SPLIT 2 ISP_VER_SPLITThe variable NumIntraSubPartitions specifies the number of transformblock subpartitions an intra luma coding block is divided into.NumIntraSubPartitions is derived as follows:

-   -   If IntraSubPartitionsSplitType is equal to ISP_NO_SPLIT,        NumIntraSubPartitions is set equal to 1.    -   Otherwise, if one of the following conditions is true,        NumIntraSubPartitions is set equal to 2:        -   cbWidth is equal to 4 and cbHeight is equal to 8,        -   cbWidth is equal to 8 and cbHeight is equal to 4.    -   Otherwise, NumIntraSubPartitions is set equal to 4.        2.3 Chroma Intra Mode Coding

For chroma intra mode coding, a total of 8 or 5 intra modes are allowedfor chroma intra mode coding depending on whether cross-component linearmodel (CCLM) is enabled or not. Those modes include five traditionalintra modes and three cross-component linear model modes. Chroma DM modeuse the corresponding luma intra prediction mode. Since separate blockpartitioning structure for luma and chroma components is enabled in Islices, one chroma block may correspond to multiple luma blocks.Therefore, for Chroma DM mode, the intra prediction mode of thecorresponding luma block covering the center position of the currentchroma block is directly inherited.

TABLE 8-2 Specification of IntraPredModeC[ xCb ][ yCb ] depending onintra_chroma_pred_mode[ xCb ][ yCb ] and IntraPredModeY[ xCb + cbWidth /2 ][ yCb + cbHeight / 2 ] when sps_cclm_enabled_flag is equal to 0IntraPredModeY[ xCb + cbWidth / 2 ][ yCb + cbHeight / 2 ]intra_chroma_pred_mode[ xCb ][ yCb ] 0 50 18 1 X ( 0 <= X <= 66 ) 0 66 00 0 0 1 50 66 50 50 50 2 18 18 66 18 18 3 1 1 1 66 1 4 (DM) 0 50 18 1 X

TABLE 8-3 Specification of IntraPredModeC[ xCb ][ yCb ] depending onintra_chroma_pred_mode[ xCb ][ yCb ] and IntraPredModeY[ xCb + cbWidth /2 ][ yCb + cbHeight / 2 ] when sps_cclm_enabled_flag is equal to 1IntraPredModeY[ xCb + cbWidth / 2 ][ yCb + cbHeight / 2 ]intra_chroma_pred_mode[ xCb ][ yCb] 0 50 18 1 X ( 0 <= X <= 66 ) 0 66 00 0 0 1 50 66 50 50 50 2 18 18 66 18 18 3 1 1 1 66 1 4 81 81 81 81 81 582 82 82 82 82 6 83 83 83 83 83 7 (DM) 0 50 18 1 X2.4 Transform Coding in VVC2.4.1 Multiple Transform Set (MTS) in VVC2.4.1.1 Explicit Multiple Transform Set (MTS)

In VTM4, large block-size transforms, up to 64×64 in size, are enabled,which is primarily useful for higher resolution video, e.g., 1080p and 4K sequences. High frequency transform coefficients are zeroed out forthe transform blocks with size (width or height, or both width andheight) equal to 64, so that only the lower-frequency coefficients areretained. For example, for an M×N transform block, with M as the blockwidth and N as the block height, when M is equal to 64, only the left 32columns of transform coefficients are kept. Similarly, when N is equalto 64, only the top 32 rows of transform coefficients are kept. Whentransform skip mode is used for a large block, the entire block is usedwithout zeroing out any values.

In addition to DCT-II which has been employed in HEVC, a MultipleTransform Selection (MTS) scheme is used for residual coding both interand intra coded blocks. It uses multiple selected transforms from theDCT8/DST7. The newly introduced transform matrices are DST-VII andDCT-VIII. The Table 4 below shows the basis functions of the selectedDST/DCT.

TABLE 4 Basis functions of transform matrices used in VVC Transform TypeBasis function T_(i)(j), i, j = 0, 1, . . . , N − 1 DCT-II${T_{i}(j)} = {\omega_{0} \cdot \sqrt{\frac{2}{N}} \cdot {\cos\left( \frac{\pi \cdot i \cdot \left( {{2j} + 1} \right)}{2N} \right)}}$${{where}\mspace{14mu}\omega_{0}} = \left\{ \begin{matrix}\sqrt{\frac{2}{N}} & {i = 0} \\1 & {i \neq 0}\end{matrix} \right.$ DCT-VIII${T_{i}(j)} = {\sqrt{\frac{4}{{2N} + 1}} \cdot {\cos\left( \frac{\pi \cdot \left( {{2i} + 1} \right) \cdot \left( {{2j} + 1} \right)}{{4N} + 2} \right)}}$DST-VII${T_{i}(j)} = {\sqrt{\frac{4}{{2N} + 1}} \cdot {\sin\left( \frac{\pi \cdot \left( {{2i} + 1} \right) \cdot \left( {j + 1} \right)}{{2N} + 1} \right)}}$

In order to keep the orthogonality of the transform matrix, thetransform matrices are quantized more accurately than the transformmatrices in HEVC. To keep the intermediate values of the transformedcoefficients within the 16-bit range, after horizontal and aftervertical transform, all the coefficients are to have 10-bit.

In order to control MTS scheme, separate enabling flags are specified atSPS level for intra and inter, respectively. When MTS is enabled at SPS,a CU level flag is signalled to indicate whether MTS is applied or not.Here, MTS is applied only for luma. The MTS CU level flag is signalledwhen the following conditions are satisfied.

-   -   Both width and height smaller than or equal to 32    -   CBF flag is equal to one

If MTS CU flag is equal to zero, then DCT2 is applied in bothdirections. However, if MTS CU flag is equal to one, then two otherflags are additionally signalled to indicate the transform type for thehorizontal and vertical directions, respectively. Transform andsignalling mapping table as shown in Table 5. When it comes to transformmatrix precision, 8-bit primary transform cores are used. Therefore, allthe transform cores used in HEVC are kept as the same, including 4-pointDCT-2 and DST-7, 8-point, 16-point and 32-point DCT-2. Also, othertransform cores including 64-point DCT-2, 4-point DCT-8, 8-point,16-point, 32-point DST-7 and DCT-8, use 8-bit primary transform cores.

TABLE 5 Mapping of decoded value of tu_mts_idx and correspondingtransform matrices for the horizontal and vertical directions. Binstring of Intra/inter tu_mts_idx tu_mts_idx Horizontal Vertical 0 0 DCT21 0 1 DST7 DST7 1 1 0 2 DCT8 DST7 1 1 1 0 3 DST7 DCT8 1 1 1 1 4 DCT8DCT8

To reduce the complexity of large size DST-7 and DCT-8, High frequencytransform coefficients are zeroed out for the DST-7 and DCT-8 blockswith size (width or height, or both width and height) equal to 32. Onlythe coefficients within the 16×16 lower-frequency region are retained.

In addition to the cases wherein different transforms are applied, VVCalso supports a mode called transform skip (TS) which is like theconcept of TS in the HEVC. TS is treated as a special case of MTS.

2.4.2 Reduced Secondary Transform (RST) Proposed in JVET-N0193

2.4.2.1 Non-Separable Secondary Transform (NSST) in JEM

In JEM, secondary transform is applied between forward primary transformand quantization (at encoder) and between de-quantization and invertprimary transform (at decoder side). As shown in FIG. 10 , 4×4 (or 8×8)secondary transform is performed depends on block size. For example, 4×4secondary transform is applied for small blocks (i.e., min (width,height)<8) and 8×8 secondary transform is applied for larger blocks(i.e., min (width, height)>4) per 8×8 block.

Application of a non-separable transform is described as follows usinginput as an example. To apply the non-separable transform, the 4×4 inputblock X

$X = \begin{bmatrix}X_{00} & X_{01} & X_{02} & X_{03} \\X_{10} & X_{11} & X_{12} & X_{13} \\X_{20} & X_{21} & X_{22} & X_{23} \\X_{30} & X_{31} & X_{32} & X_{33}\end{bmatrix}$

is first represented as a vector

:

$\overset{\rightharpoonup}{X} = \left\lbrack {X_{00}\ X_{01}\ X_{02}\ X_{03}\ X_{10}\ X_{11}\ X_{12}\ X_{13}\ X_{20}\ X_{21}\ X_{22}\ X_{23}\ X_{30}\ X_{31}\ X_{32}\ X_{33}} \right\rbrack^{T}$

The non-separable transform is calculated as

=T·

, where

indicates the transform coefficient vector, and Tis a 16×16 transformmatrix. The 16×1 coefficient vector

is subsequently re-organized as 4×4 block using the scanning order forthat block (horizontal, vertical or diagonal). The coefficients withsmaller index will be placed with the smaller scanning index in the 4×4coefficient block. There are totally 35 transform sets and 3non-separable transform matrices (kernels) per transform set are used.The mapping from the intra prediction mode to the transform set ispre-defined. For each transform set, the selected non-separablesecondary transform (NSST) candidate is further specified by theexplicitly signalled secondary transform index. The index is signalledin a bit-stream once per Intra CU after transform coefficients.

2.4.2.2 Reduced Secondary Transform (RST) in JVET-N0193

The RST (a.k.a. Low Frequency Non-Separable Transform (LFNST)) wasintroduced in JVET-K0099 and 4 transform set (instead of 35 transformsets) mapping introduced in JVET-L0133. In this JVET-N0193, 16×64(further reduced to 16×48) and 16×16 matrices are employed. Fornotational convenience, the 16×64 (reduced to 16×48) transform isdenoted as RST8×8 and the 16×16 one as RST4×4. FIG. 11 shows an exampleof RST.

2.4.2.2.1 RST Computation

The main idea of a Reduced Transform (RT) is to map an N dimensionalvector to an R dimensional vector in a different space, where R/N(R<N)is the reduction factor.

The RT matrix is an R×N matrix as follows:

$T_{R \times N} = \begin{bmatrix}t_{11} & t_{12} & t_{13} & \ldots & t_{1N} \\t_{21} & t_{22} & t_{23} & & t_{2N} \\ & \vdots & & \ddots & \vdots \\t_{R1} & t_{R2} & t_{R3} & \ldots & t_{RN}\end{bmatrix}$

where the R rows of the transform are R bases of the N dimensionalspace. The invert transform matrix for RT is the transpose of itsforward transform. The forward and invert RT are depicted in FIG. 12 .

In this contribution, the RST8×8 with a reduction factor of 4 (¼ size)is applied. Hence, instead of 64×64, which is conventional 8×8non-separable transform matrix size, 16×64 direct matrix is used. Inother words, the 64×16 invert RST matrix is used at the decoder side togenerate core (primary) transform coefficients in 8×8 top-left regions.The forward RST8×8 uses 16×64 (or 8×64 for 8×8 block) matrices so thatit produces non-zero coefficients only in the top-left 4×4 region withinthe given 8×8 region. In other words, if RST is applied then the 8×8region except the top-left 4×4 region will have only zero coefficients.For RST4×4, 16×16 (or 8×16 for 4×4 block) direct matrix multiplicationis applied.

An invert RST is conditionally applied when the following two conditionsare satisfied:

-   -   Block size is greater than or equal to the given threshold (W>=4        && H>=4)    -   Transform skip mode flag is equal to zero

If both width (W) and height (H) of a transform coefficient block isgreater than 4, then the RST8×8 is applied to the top-left 8×8 region ofthe transform coefficient block. Otherwise, the RST4×4 is applied on thetop-left min(8, W)×min(8, H) region of the transform coefficient block.

If RST index is equal to 0, RST is not applied. Otherwise, RST isapplied, of which kernel is chosen with the RST index. The RST selectionmethod and coding of the RST index are explained later.

Furthermore, RST is applied for intra CU in both intra and inter slices,and for both Luma and Chroma. If a dual tree is enabled, RST indices forLuma and Chroma are signaled separately. For inter slice (the dual treeis disabled), a single RST index is signaled and used for both Luma andChroma.

2.4.2.2.2 Restriction of RST

When ISP mode is selected, RST is disabled, and RST index is notsignaled, because performance improvement was marginal even if RST isapplied to every feasible partition block. Furthermore, disabling RSTfor ISP-predicted residual could reduce encoding complexity.

2.4.2.2.3 RST Selection

A RST matrix is chosen from four transform sets, each of which consistsof two transforms. Which transform set is applied is determined fromintra prediction mode as the following:

-   -   (1) If one of three CCLM modes is indicated, transform set 0 is        selected.    -   (2) Otherwise, transform set selection is performed according to        the following table:

The transform set selection table Tr. set IntraPredMode indexIntraPredMode < 0 1 0 <= IntraPredMode <= 1 0  2 <= IntraPredMode <= 121 13 <= IntraPredMode <= 23 2 24 <= IntraPredMode <= 44 3 45 <=IntraPredMode <= 55 2 56 <= IntraPredMode 1

The index to access the above table, denoted as IntraPredMode, have arange of [−14, 83], which is a transformed mode index used for wideangle intra prediction.

Later, the Low Frequency Non-Separable Transform (LFNST, a.k.a. RST) setselection for chroma blocks coded in CCLM modes is modified to be basedon a variable IntraPredMode_CCLM, wherein the IntraPredMode_CCLM has arange of [−14, 80]. The IntraPredMode_CCLM is determined by theco-located luma intra prediction mode and the dimension of the currentchroma block.

When dual tree is enabled, the block (e.g., PU) covering thecorresponding luma sample of the top-left chroma sample in the currentchroma block is defined as the co-located luma block. An example wasshown in FIGS. 24 a and 24 b with the co-located position denoted by TL

2.4.2.2.4 RST Matrices of Reduced Dimension

As a further simplification, 16×48 matrices are applied instead of 16×64with the same transform set configuration, each of which takes 48 inputdata from three 4×4 blocks in a top-left 8×8 block excludingright-bottom 4×4 block (as shown in FIG. 13 ).

2.4.2.2.5 RST Signaling

The forward RST8×8 uses 16×48 matrices so that it produces non-zerocoefficients only in the top-left 4×4 region within the first 3 4×4region. In other words, if RST8×8 is applied, only the top-left 4×4 (dueto RST8×8) and bottom right 4×4 region (due to primary transform) mayhave non-zero coefficients. As a result, RST index is not coded when anynon-zero element is detected within the top-right 4×4 and bottom-left4×4 block region (shown in FIG. 14 , and referred to as “zero-out”regions) because it implies that RST was not applied. In such a case,RST index is inferred to be zero.

2.4.2.2.6 Zero-Out Region within One CG

Usually, before applying the invert RST on a 4×4 sub-block, anycoefficient in the 4×4 sub-block may be non-zero. However, it isconstrained that in some cases, some coefficients in the 4×4 sub-blockmust be zero before invert RST is applied on the sub-block.

Let nonZeroSize be a variable. It is required that any coefficient withthe index no smaller than nonZeroSize when it is rearranged into a 1-Darray before the invert RST must be zero.

When nonZeroSize is equal to 16, there is no zero-out constrain on thecoefficients in the top-left 4×4 sub-block.

In JVET-N0193, when the current block size is 4×4 or 8×8, nonZero Sizeis set equal to 8 (that is, coefficients with the scanning index in therange [8, 15] as show in FIG. 14 , shall be 0). For other blockdimensions, nonZeroSize is set equal to 16.

2.4.2.2.7 Description of RST in Working Draft

7.3.2.3 Sequence Parameter Set RBSP Syntax

Descriptor seq_parameter_set_rbsp( ) {  ......  sps_mts_enabled_flagu(1)  if( sps_mts_enabled_flag ) {   sps_explicit_mts_intra_enabled_flagu(1)   sps_explicit_mts_inter_enabled_flag u(1)  }  ... sps_st_enabled_flag u(1)  ... }7.3.7.11 Residual Coding Syntax

Descriptor residual_coding( x0, y0, log2TbWidth, log2TbHeight, cIdx ) {...  if( coded_sub_block_flag[ xS ][ yS ] && ( n > 0 | |!inferSbDcSigCoeffFlag ) &&   ( xC != LastSignificantCoeffX | | yC !=Last SignificantCoeffY ) ) {   sig_coeff_flag[ xC ][ yC ] ae(v)  remBinsPass1−−   if( sig_coeff_flag[ xC ][ yC ] )   inferSbDcSigCoeffFlag = 0  }  if( sig_coeff_flag[ xC ][ yC ] ) {  if( !transform_skip_flag[ x0 ][ y0 ] ) {   numSigCoeff++    if( ( ( (log2TbWidth == 2 && log2TbHeight == 2 ) | | ( log2TbWidth == 3 &&log2TbHeight == 3 ) ) && n >= 8 && i == 0 ) | | ( ( log2TbWidth >= 3 &&log2TbHeight >= 3 && ( i == 1 | | i == 2 ) ) ) ) {    numZeroOutSigCoeff++    }   }   abs_level_gt1_flag[ n ] ae(v) ...7.3.7.5 Coding Unit Syntax

Descriptor coding_unit( x0, y0, cbWidth, cbHeight, treeType ) { ...  if(!pcm_flag[ x0 ][ y0 ] ) {   if( CuPredMode[ x0 ][ y0 ] != MODE_INTRA &&merge_flag[ x0 ][ y0 ] = = 0 )    cu_cbf ae(v)   if( cu_cbf ) {    if(CuPredMode[ x0 ][ y0 ] = = MODE_INTER && sps_sbt_enabled_flag &&    !ciip_flag[ x0 ][ y0 ] ) {     if( cbWidth <= MaxSbtSize && cbHeight<= MaxSbtSize ) {      allowSbtVerH = cbWidth >= 8      allowSbtVerQ =cbWidth >= 16      allowSbtHorH = cbHeight >= 8      allowSbtHorQ =cbHeight >= 16      if( allowSbtVerH | | allowSbtHorH | | allowSbtVerQ || allowSbtHorQ )       cu_sbt_flag ae(v)     }     if( cu_sbt_flag ) {     if( ( allowSbtVerH | | allowSbtHorH ) && ( allowSbtVerQ | |allowSbtHorQ) )       cu_sbt_quad_flag ae(v)      if( ( cu_sbt_quad_flag&& allowSbtVerQ && allowSbtHorQ ) | |        ( !cu_sbt_quad_flag &&allowSbtVerH && allowSbtHorH ) )       cu_sbt_horizontal_flag ae(v)     cu_sbt_pos_flag ae(v)     }    }    numZeroOutSigCoeff = 0   transform_tree( x0, y0, cbWidth, cbHeight, treeType )    if( Min(cbWidth, cbHeight ) >= 4 && sps_st_enabled_flag == 1 && CuPredMode[ x0][ y0 ] = = MODE_INTRA && IntraSubPartitionsSplitType == ISP_NO_SPLIT ){     if( ( numSigCoeff > ( ( treeType == SINGLE_TREE ) ? 2 : 1 ) ) &&numZeroOutSigCoeff == 0 ) {       st_idx[ x0 ][ y0 ] ae(v)     }    }  }  } }sps_st_enabled_flag equal to 1 specifies that st_idx may be present inthe residual coding syntax for intra coding units. sps_st_enabled_flagequal to 0 specifies that st_idx is not present in the residual codingsyntax for intra coding units.st_idx[x0] [y0] specifies which secondary transform kernel is appliedbetween two candidate kernels in a selected transform set. st_idx[x0][y0] equal to 0 specifies that the secondary transform is not applied.The array indices x0, y0 specify the location (x0, y0) of the top-leftsample of the considered transform block relative to the top-left sampleof the picture.When st_idx[x0] [y0] is not present, st_idx[x0] [y0] is inferred to beequal to 0.It is noted that whether to send the st_idx is dependent on number ofnon-zero coefficients in all TUs within a CU (e.g., for single tree,number of non-zero coefficients in 3 blocks (i.e., Y, Cb, Cr); for dualtree and luma is coded, number of non-zero coefficients in the lumablock; for dual tree and chroma is coded, number of non-zerocoefficients in the two chroma blocks). In addition, the threshold isdependent on the partitioning structure, (treeType=SINGLE_TREE)? 2:1).Bins of st_idx are context-coded. More specifically, the followingapplies:

TABLE 9-9 Syntax elements and associated binarizations BinarizationSyntax element Process Input parameters Syntax . . . . . . . . .structure st_idx[ ][ ] TR cMax = 2, cRiceParam = 0

TABLE 9-15 Assignment of ctxInc to syntax elements with context codedbins Syntax binIdx element 0 1 2 3 4 >=5 . . . . . . . . . . . . . . . .. . . . . st_idx[ ][ ] 0, 1, 4, 5 2, 3, 6, 7 na na na na (clause9.5.4.2.8) (clause 9.5.4.2.8) . . . . . . . . . . . . . . . . . . . . .9.5.4.2.8 Derivation Process of ctxInc for the Syntax Element st_idxInputs to this process are the colour componentindex cIdx, the luma orchroma location (x0, y0) specifying the top-left sample of the currentluma or chromacoding block relative to the top-left sample of thecurrent picture depending on cIdx, the tree type tree Type, the lumaintra prediction mode IntraPredModeY[x0] [y0] as specified in clause8.4.2, the syntax element intra_chroma_pred_mode[x0] [y0] specifying theintra prediction mode for chroma samples as specified in clause 7.4.7.5,and the multiple transform selection index tu_mts_idx[x0] [y0].Output of this process is the variable ctxInc.The variable intraModeCtx is derived as follows:If cIdx is equal to 0, intraModeCtx is derived as follows:intraModeCtx=(IntraPredModeY[x0][y0]<=1)?1:0Otherwise (cIdx is greater than 0), intraModeCtx is derived as follows:intraModeCtx=(intra_chroma_pred_mode[x0][y0]>=4)?1:0The variable mtsCtx is derived as follows:mtsCtx=(tu_mts_idx[x0][y0]==0&& treeType!=SINGLE_TREE)?1:0The variable ctxInc is derived as follows:ctxInc=(binIdx<<1)+intraModeCtx+(mtsCtx<<2)2.4.2.2.8 Summary of RST UsageRST may be enabled only when the number of non-zero coefficients in oneblock is greater than 2 and 1 for sing and separate tree, respectively.In addition, the following restrictions of locations of non-zerocoefficients for RST applied Coding Groups (CGs) is also required whenRST is enabled.

TABLE 1 Usage of RST Potential locations Which CG that of non-zerocoeffs RST applied in the CGs RST # of CGs that to may have applied to(nonZeroSize Block size RST type RST applied to non-zero coeffs relativeto one CG) 4 × 4 RST4 × 4 1 (Top-left 4 × 4) Top-left 4 × 4 First 8 indiagonal (16 × 16) scan order (0 . . . 7 in FIG. 16: diagonal up-rightscan order (4 × 4 as a CG for example), nonZeroSize = 8 4 × 8/8 × 4 RST4× 4 1 (Top-left 4 × 4) Top-left 4 × 4 all, nonZeroSize = 16 (16 × 16) 4× N and N × 4 RST4 × 4 2 (4 × N: up most 4 × 8; 4 × N: up most 4 × 8;all, nonZeroSize = 16 (N > 8) (16 × 16) N × 4: left most 4 × 8) N × 4:left most 4 × 8 8 × 8 RST8 × 8 3 (with only 1 CG Top-left 4 × 4 First 8in diagonal (16 × 48) may have non-zero scan order coeffs after (0 . . .7 in FIG. 16: forward RST) diagonal up-right scan order (4 × 4 as a CGfor example), nonZeroSize = 8 Others RST8 × 8 3 (with only 1 CG Top-left4 × 4 all, nonZeroSize = 16 (W*H, (16 × 48) may have non-zero W > 8, H >8) coeffs after forward RST)2.4.3 Sub-Block Transform

For an inter-predicted CU with cu_cbf equal to 1, cu_sbt_flag may besignaled to indicate whether the whole residual block or a sub-part ofthe residual block is decoded. In the former case, inter MTS informationis further parsed to determine the transform type of the CU. In thelatter case, a part of the residual block is coded with inferredadaptive transform and the other part of the residual block is zeroedout. The SBT is not applied to the combined inter-intra mode.

In sub-block transform, position-dependent transform is applied on lumatransform blocks in SBT-V and SBT-H (chroma TB always using DCT-2). Thetwo positions of SBT-H and SBT-V are associated with different coretransforms. More specifically, the horizontal and vertical transformsfor each SBT position is specified in FIG. 3 . For example, thehorizontal and vertical transforms for SBT-V position 0 is DCT-8 andDST-7, respectively. When one side of the residual TU is greater than32, the corresponding transform is set as DCT-2. Therefore, thesub-block transform jointly specifies the TU tiling, cbf, and horizontaland vertical transforms of a residual block, which may be considered asyntax shortcut for the cases that the major residual of a block is atone side of the block.

2.4.3.1 Syntax Elements

7.3.7.5 Coding Unit Syntax

Descriptor coding_unit( x0, y0, cbWidth, cbHeight, treeType ) {  if(slice_type != I || sps_ibc_enabled_flag ) {   if( treeType !=DUAL_TREE_CHROMA )    cu_skip_flag[ x0 ][ y0 ] ae(v)   if( cu_skip_flag[x0 ][ y0 ] = = 0 && slice_type != I )    pred_mode_flag ae(v)   if( ( (slice_type = = I && cu_skip_flag[ x0 ][ y0 ] = =0 ) ||    ( slice_type!= I && CuPredMode[ x0 ][ y0 ] != MODE_INTRA ) ) &&   sps_ibc_enabled_flag )    pred_mode_ibc_flag ae(v)  }  if(CuPredMode[ x0 ][ y0 ] = = MODE_INTRA ) { ...  } else if( treeType !=DUAL_TREE_CHROMA ) { /* MODE_INTER or MODE_IBC */ ...  }  if( !pcm_flag[x0 ][ y0 ] ) {   if( CuPredMode[ x0 ][ y0 ] != MODE_INTRA && merge_flag[x0 ][ y0 ] = = 0 )    cu_cbf ae(v)   if( cu_cbf) {    if( CuPredMode[ x0][ y0 ] = = MODE_INTER && sps_sbt_enabled_flag &&     !ciip_flag[ x0 ][y0 ] ) {     if( cbWidth <= MaxSbtSize && cbHeight <= MaxSbtSize ) {     allowSbtVerH = cbWidth >= 8      allowSbtVerQ = cbWidth >= 16     allowSbtHorH = cbHeight >= 8      allowSbtHorQ = cbHeight >= 16     if( allowSbtVerH || allowSbtHorH || allowSbtVerQ || allowSbtHorQ )      cu_sbt_flag ae(v)     }     if( cu_sbt_flag ) {      if( (allowSbtVerH || allowSbtHorH ) && ( allowSbtVerQ || allowSbtHorQ) )      cu_sbt_quad_flag ae(v)      if( ( cu_sbt_quad_flag && allowSbtVerQ&& allowSbtHorQ ) ||        ( !cu_sbt_quad_flag && allowSbtVerH &&allowSbtHorH ) )       cu_sbt_horizontal_flag ae(v)      cu_sbt_pos_flagae(v)     }    }    transform_tree( x0, y0, cbWidth, cbHeight, treeType)   }  } }cu_sbt_flag equal to 1 specifies that for the current coding unit,subblock transform is used. cu_sbt_flag equal to 0 specifies that forthe current coding unit, subblock transform is not used.When cu_sbt_flag is not present, its value is inferred to be equal to 0.

-   -   NOTE—: When subblock transform is used, a coding unit is split        into two transform units; one transform unit has residual data,        the other does not have residual data.        cu_sbt_quad_flag equal to 1 specifies that for the current        coding unit, the subblock transform includes a transform unit of        ¼ size of the current coding unit. cu_sbt_quad_flag equal to 0        specifies that for the current coding unit the subblock        transform includes a transform unit of ½ size of the current        coding unit.        When cu_sbt_quad_flag is not present, its value is inferred to        be equal to 0.        cu_sbt_horizontal_flag equal to 1 specifies that the current        coding unit is split horizontally into 2 transform units.        cu_sbt_horizontal_flag[x0] [y0] equal to 0 specifies that the        current coding unit is split vertically into 2 transform units.        When cu_sbt_horizontal_flag is not present, its value is derived        as follows:    -   If cu_sbt_quad_flag is equal to 1, cu_sbt_horizontal_flag is set        to be equal to allow SbtHorQ.    -   Otherwise (cu_sbtquad_flag is equal to 0),        cu_sbt_horizontal_flag is set to be equal to allow SbtHorH.        cu_sbt_pos_flag equal to 1 specifies that the tu_cbf_luma,        tu_cbf_cb and tu_cbf_cr of the first transform unit in the        current coding unit are not present in the bitstream.        cu_sbt_pos_flag equal to 0 specifies that the tu_cbf_luma,        tu_cbf_cb and tu_cbf_cr of the second transform unit in the        current coding unit are not present in the bitstream.        The variable SbtNumFourthsTb0 is derived as follows:        sbtMinNumFourths=cu_sbt_quad_flag?1:2  (7-117)        SbtNumFourthsTb0=cu_sbt_pos_flag?(4−sbtMinNumFourths):sbtMinNumFourths  (7-118)        sps_sbt_max_size_64_flag equal to 0 specifies that the maximum        CU width and height for allowing subblock transform is 32 lum a        samples. sps_sbt_max_size_64_flag equal to 1 specifies that the        maximum CU width and height for allowing subblock transform is        64 lum a samples.        MaxSbtSize=sps_sbt_max_size_64_flag?64:32  (7-33)        2.4.4 Quantized Residual Domain Block Differential Pulse-Code        Modulation Coding (QR-BDPCM)

In JVET-N0413, quantized residual domain BDPCM (denote as RBDPCMhereinafter) is proposed. The intra prediction is done on the entireblock by sample copying in prediction direction (horizontal or verticalprediction) similar to intra prediction. The residual is quantized andthe delta between the quantized residual and its predictor (horizontalor vertical) quantized value is coded.

For a block of size M (rows)×N (cols), let r_(i,j), 0≤i≤M−1, 0≤j≤N−1. bethe prediction residual after performing intra prediction horizontally(copying left neighbor pixel value across the predicted block line byline) or vertically (copying top neighbor line to each line in thepredicted block) using unfiltered samples from above or left blockboundary samples. Let Q(r_(i,j)), 0≤i≤M−1, 0≤j≤N−1 denote the quantizedversion of the residual r_(i,j), where residual is difference betweenoriginal block and the predicted block values. Then the block DPCM isapplied to the quantized residual samples, resulting in modified M×Narray {tilde over (R)} with elements {tilde over (r)}_(i,j). Whenvertical BDPCM is signaled:

${\overset{˜}{r}}_{i,j} = \left\{ \begin{matrix}{{Q\left( r_{i,j} \right)}\ ,\ {i = 0},\ {0 \leq j \leq \left( {N - 1} \right)}} \\{{{Q\left( r_{i,j} \right)} - {Q\left( r_{{({i - 1})},j} \right)}}\ ,\ {1 \leq i \leq \left( {M - 1} \right)}\ ,\ {0 \leq j \leq \left( {N - 1} \right)}}\end{matrix} \right.$

For horizontal prediction, similar rules apply, and the residualquantized samples are obtained by

${\overset{˜}{r}}_{i,j} = \left\{ \begin{matrix}{{Q\left( r_{i,j} \right)}\ ,\ {0 \leq i \leq \left( {M - 1} \right)},\ {j = 0}} \\{{{Q\left( r_{i,j} \right)} - {Q\left( r_{i,{({j - 1})}} \right)}}\ ,\ {0 \leq i \leq \left( {M - 1} \right)}\ ,\ {1 \leq j \leq \left( {N - 1} \right)}}\end{matrix} \right.$

The residual quantized samples {tilde over (r)}_(i,j) are sent to thedecoder.

On the decoder side, the above calculations are reversed to produceQ(r_(i,j)), 0≤i≤M−1, 0≤j≤N−1. For vertical prediction case,

${{Q\left( r_{i,j} \right)} = {\sum\limits_{k = 0}^{i}{\overset{˜}{r}}_{k,j}}},{0 \leq i \leq \left( {M - 1} \right)},{0 \leq j \leq \left( {N - 1} \right)}$

For horizontal case,

${{Q\left( r_{i,j} \right)} = {\sum\limits_{k = 0}^{j}{\overset{˜}{r}}_{i,k}}},{0 \leq i \leq \left( {M - 1} \right)},{0 \leq j \leq \left( {N - 1} \right)}$

The invert quantized residuals, Q⁻¹ (Q(r_(i,j))), are added to the intrablock prediction values to produce the reconstructed sample values.

When QR-BDPCM is selected, there is no transform applied.

2.5 Entropy Coding of Coefficients

2.5.1 Coefficients Coding of Transform-Applied Blocks

In HEVC, transform coefficients of a coding block are coded usingnon-overlapped coefficient groups (or subblocks), and each CG containsthe coefficients of a 4×4 block of a coding block. The CGs inside acoding block, and the transform coefficients within a CG, are codedaccording to pre-defined scan orders.

The CGs inside a coding block, and the transform coefficients within aCG, are coded according to pre-defined scan orders. Both CG andcoefficients within a CG follows the diagonal up-right scan order. Anexample for 4×4 block and 8×8 scanning order is depicted in FIG. 16 andFIG. 17 , respectively.

Note that the coding order is the reversed scanning order (i.e.,decoding from CG3 to CG0 in FIG. 17 ), when decoding one block, the lastnon-zero coefficient's coordinate is firstly decoded.

The coding of transform coefficient levels of a CG with at least onenon-zero transform coefficient may be separated into multiple scanpasses. In the first pass, the first bin (denoted by bin0, also referredas significant_coeff_flag, which indicates the magnitude of thecoefficient is larger than 0) is coded. Next, two scan passes forcontext coding the second/third bins (denoted by bin1 and bin2,respectively, also referred as coeff_abs_greater1_flag andcoeff_abs_greater2_flag) may be applied. Finally, two more scan passesfor coding the sign information and the remaining values (also referredas coeff_abs_level_remaining) of coefficient levels are invoked, ifnecessary. Note that only bins in the first three scan passes are codedin a regular mode and those bins are termed regular bins in thefollowing descriptions.

In the VVC 3, for each CG, the regular coded bins and the bypass codedbins are separated in coding order; first all regular coded bins for asubblock are transmitted and, thereafter, the bypass coded bins aretransmitted. The transform coefficient levels of a subblock are coded infive passes over the scan positions as follows:

-   -   Pass 1: coding of significance (sig_flag), greater 1 flag        (gt1_flag), parity (par_level_flag) and greater 2 flags        (gt2_flag) is processed in coding order. If sig_flag is equal to        1, first the gt1_flag is coded (which specifies whether the        absolute level is greater than 1). If gt1_flag is equal to 1,        the par_flag is additionally coded (it specifies the parity of        the absolute level minus 2).    -   Pass 2: coding of remaining absolute level (remainder) is        processed for all scan positions with gt2_flag equal to 1 or        gt1_flag equal to 1. The non-binary syntax element is binarized        with Golomb-Rice code and the resulting bins are coded in the        bypass mode of the arithmetic coding engine.    -   Pass 3: absolute level (absLevel) of the coefficients for which        no sig_flag is coded in the first pass (due to reaching the        limit of regular-coded bins) are completely coded in the bypass        mode of the arithmetic coding engine using a Golomb-Rice code.    -   Pass 4: coding of the signs (sign_flag) for all scan positions        with sig_coeff_flag equal to 1

It is guaranteed that no more than 32 regular-coded bins (sig_flag,par_flag, gt1_flag and gt2_flag) are encoded or decoded for a 4×4subblock. For 2×2 chroma subblocks, the number of regular-coded bins islimited to 8.

The Rice parameter (ricePar) for coding the non-binary syntax elementremainder (in Pass 3) is derived similar to HEVC. At the start of eachsubblock, ricePar is set equal to 0. After coding a syntax elementremainder, the Rice parameter is modified according to predefinedequation. For coding the non-binary syntax element absLevel (in Pass 4),the sum of absolute values sumAbs in a local template is determined. Thevariables ricePar and posZero are determined based on dependentquantization and sumAbs by a table look-up. The intermediate variablecodeValue is derived as follows:

-   -   If absLevel[k] is equal to 0, codeValue is set equal to posZero;    -   Otherwise, if absLevel[k] is less than or equal to posZero,        codeValue is set equal to absLevel[k]−1;    -   Otherwise (absLevel[k] is greater than posZero), codeValue is        set equal to absLevel[k].

The value of codeValue is coded using a Golomb-Rice code with Riceparameter ricePar.

2.5.1.1 Context Modeling for Coefficient Coding

The selection of probability models for the syntax elements related toabsolute values of transform coefficient levels depends on the values ofthe absolute levels or partially reconstructed absolute levels in alocal neighbourhood. The template used is illustrated in FIG. 18 .

The selected probability models depend on the sum of the absolute levels(or partially reconstructed absolute levels) in a local neighborhood andthe number of absolute levels greater than 0 (given by the number ofsig_coeff_flags equal to 1) in the local neighborhood. The contextmodelling and binarization depends on the following measures for thelocal neighborhood:

-   -   numSig: the number of non-zero levels in the local neighborhood;    -   sumAbs1: the sum of partially reconstructed absolute levels        (absLevel1) after the first pass in the local neighborhood;    -   sumAbs: the sum of reconstructed absolute levels in the local        neighborhood    -   diagonal position (d): the sum of the horizontal and vertical        coordinates of a current scan position inside the transform        block

Based on the values of numSig, sumAbs1, and d, the probability modelsfor coding sig_flag, par_flag, gt1_flag, and gt2_flag are selected. TheRice parameter for binarizing abs_remainder is selected based on thevalues of sumAbs and numSig.

2.5.1.2 Dependent Quantization (DQ)

In addition, the same HEVC scalar quantization is used with a newconcept called dependent scale quantization. Dependent scalarquantization refers to an approach in which the set of admissiblereconstruction values for a transform coefficient depends on the valuesof the transform coefficient levels that precede the current transformcoefficient level in reconstruction order. The main effect of thisapproach is that, in comparison to conventional independent scalarquantization as used in HEVC, the admissible reconstruction vectors arepacked denser in the N-dimensional vector space (N represents the numberof transform coefficients in a transform block). That means, for a givenaverage number of admissible reconstruction vectors per N-dimensionalunit volume, the average distortion between an input vector and theclosest reconstruction vector is reduced. The approach of dependentscalar quantization is realized by: (a) defining two scalar quantizerswith different reconstruction levels and (b) defining a process forswitching between the two scalar quantizers.

The two scalar quantizers used, denoted by Q0 and Q1, are illustrated inFIG. 19 . The location of the available reconstruction levels isuniquely specified by a quantization step size A. The scalar quantizerused (Q0 or Q1) is not explicitly signalled in the bitstream. Instead,the quantizer used for a current transform coefficient is determined bythe parities of the transform coefficient levels that precede thecurrent transform coefficient in coding/reconstruction order.

As illustrated in FIG. 20 , the switching between the two scalarquantizers (Q0 and Q1) is realized via a state machine with four states.The state can take four different values: 0, 1, 2, 3. It is uniquelydetermined by the parities of the transform coefficient levels precedingthe current transform coefficient in coding/reconstruction order. At thestart of the inverse quantization for a transform block, the state isset equal to 0. The transform coefficients are reconstructed in scanningorder (i.e., in the same order they are entropy decoded). After acurrent transform coefficient is reconstructed, the state is updated asshown in FIG. 20 , where k denotes the value of the transformcoefficient level.

2.5.1.3 Syntax and Semantics

7.3.7.11 Residual Coding Syntax

Descriptor residual_coding( x0, y0, log2TbWidth, log2TbHeight, cIdx ) { if( ( tu_mts_idx[ x0 ][ y0 ] > 0 ||    ( cu_sbt_flag && log2TbWidth < 6&& log2TbHeight < 6 ) )    && cIdx = = 0 && log2TbWidth > 4 )  log2TbWidth = 4  else   log2TbWidth = Min( log2TbWidth, 5 )  if(tu_mts_idx[ x0 ][ y0 ] > 0 ||    ( cu_sbt_flag && log2TbWidth < 6 &&log2TbHeight < 6 ) )    && cIdx = = 0 && log2TbHeight > 4 )  log2TbHeight = 4  else   log2TbHeight = Min( log2TbHeight, 5 )  if(log2TbWidth > 0 )   last_sig_coeff_x_prefix ae(v)  if( log2TbHeight > 0)   last_sig_coeff_y_prefix ae(v)  if( last_sig_coeff_x_prefix > 3 )  last_sig_coeff_x_suffix ae(v)  if( last_sig_coeff_y_prefix > 3 )  last_sig_coeff_y_suffix ae(v)  log2SbW = ( Min( log2TbWidth,log2TbHeight ) < 2 ? 1 : 2 )  log2SbH = log2SbW  if ( log2TbWidth < 2 &&cIdx = = 0 ) {   log2SbW = log2TbWidth   log2SbH = 4 − log2SbW  } elseif ( log2TbHeight < 2 && cIdx = = 0 ) {   log2SbH = log2TbHeight  log2SbW = 4 − log2SbH  }  numSbCoeff = 1 << ( log2SbW + log2SbH ) lastScanPos = numSbCoeff  lastSubBlock = ( 1 << ( log2TbWidth +log2TbHeight − ( log2SbW + log2SbH ) ) ) − 1  do {   if( lastScanPos = =0 ) {    lastScanPos = numSbCoeff    lastSubBlock− −   }   lastScanPos−−   xS = DiagScanOrder[ log2TbWidth − log2SbW ][ log2TbHeight − log2SbH]         [ lastSubBlock ][ 0 ]   yS = DiagScanOrder[ log2TbWidth −log2SbW ][ log2TbHeight − log2SbH ]         [ lastSubBlock ][ 1 ]   xC =( xS << log2SbW ) + DiagScanOrder[ log2SbW ][ log2SbH ][ lastScanPos ][0 ]   yC = ( yS << log2SbH ) + DiagScanOrder[ log2SbW ][ log2SbH ][lastScanPos ][ 1 ]  } while( ( xC != LastSignificantCoeffX ) || ( yC !=LastSignificantCoeffY ) )  QState = 0  for( i = lastSubBlock; i >= 0; i−− ) {   startQStateSb = QState   xS = DiagScanOrder[ log2TbWidth −log2SbW ][ log2TbHeight − log2SbH ]         [ lastSubBlock ][ 0 ]   yS =DiagScanOrder[ log2TbWidth − log2SbW ][ log2TbHeight − log2SbH ]        [ lastSubBlock ][ 1 ]   inferSbDcSigCoeffFlag = 0   if( ( i <lastSubBlock ) && ( i > 0 ) ) {    coded_sub_block_flag[ xS ][ yS ]ae(v)    inferSbDcSigCoeffFlag = 1   }   firstSigScanPosSb = numSbCoeff  lastSigScanPosSb = −1   remBinsPass1 = ( ( log2SbW + log2SbH ) < 4 ? 8: 32 )   firstPosMode0 = ( i = = lastSubBlock ? lastScanPos : numSbCoeff− 1 )   firstPosMode1 = −1   for( n = firstPosMode0; n >= 0 &&remBinsPass1 >= 4; n− − ) {    xC = ( xS << log2SbW ) + DiagScanOrder[log2SbW ][ log2SbH ][ n ][ 0 ]    yC = ( yS << log2SbH ) +DiagScanOrder[ log2SbW ][ log2SbH ][ n ][ 1 ]    if(coded_sub_block_flag[ xS ][ yS ] && ( n > 0 || !inferSbDcSigCoeffFlag )&&     ( xC != LastSignificantCoeffX || yC != Last SignificantCoeffY ) ){     sig_coeff_flag[ xC ][ yC ] ae(v)     remBinsPass1− −     if(sig_coeff_flag[ xC ][ yC ] )      inferSbDcSigCoeffFlag = 0    }    if(sig_coeff_flag[ xC ][ yC ] ) {     abs_level_gt1_flag[ n ] ae(v)    remBinsPass1− −     if( abs_level_gt1_flag[ n ] ) {     par_level_flag[ n ] ae(v)      remBinsPass1− −     abs_level_gt3_flag[ n ] ae(v)      remBinsPass1− −     }     if(lastSigScanPosSb = = −1 )      lastSigScanPosSb = n    firstSigScanPosSb = n    }    AbsLevelPass1[ xC ][ yC ] =sig_coeff_flag[ xC ][ yC ] + par_level_flag[ n ] +            abs_level_gt1_flag[ n ] + 2 * abs_level_gt3_flag[ n ]    if(dep_quant_enabled_flag )     QState = QStateTransTable[ QState ][AbsLevelPass1[ xC ][ yC ] & 1 ]    if( remBinsPass1 < 4 )    firstPosMode1 = n − 1   }   for( n = numSbCoeff − 1; n >=firstPosMode1; n− − ) {    xC = ( xS << log2SbW ) + DiagScanOrder[log2SbW ][ log2SbH ][ n ][ 0 ]    yC = ( yS << log2SbH ) +DiagScanOrder[ log2SbW ][ log2SbH ][ n ][ 1 ]    if( abs_level_gt3_flag[n ] )     abs_remainder[ n ] ae(v)    AbsLevel[ xC ][ yC ] =AbsLevelPass1[ xC ][ yC ] +2 * abs_remainder[ n ]   }   for( n =firstPosMode1; n >= 0; n− − ) {    xC = ( xS << log2SbW ) +DiagScanOrder[ log2SbW ][ log2SbH ][ n ][ 0 ]    yC = ( yS << log2SbH) + DiagScanOrder[ log2SbW ][ log2SbH ][ n ][ 1 ]    dec_abs_level[ n ]ae(v)    if(AbsLevel[ xC ][ yC ] > 0 )     firstSigScanPosSb = n    if(dep_quant_enabled_flag )     QState = QStateTransTable[ QState ][AbsLevel[ xC ][ yC ] & 1 ]   }   if( dep_quant_enabled_flag ||!sign_data_hiding_enabled_flag )    signHidden = 0   else    signHidden= ( lastSigScanPosSb − firstSigScanPosSb > 3 ? 1 : 0 )   for( n =numSbCoeff − 1; n >= 0; n− − ) {    xC = ( xS << log2SbW ) +DiagScanOrder[ log2SbW ][ log2SbH ][ n ][ 0 ]    yC = ( yS << log2SbH) + DiagScanOrder[ log2SbW ][ log2SbH ][ n ][ 1 ]    if( ( AbsLevel[ xC][ yC ] > 0 ) &&     ( !signHidden || ( n != firstSigScanPosSb ) ) )    coeff_sign_flag[ n ] ae(v)   }   if( dep_quant_enabled_flag ) {   QState = startQStateSb    for( n = numSbCoeff − 1; n >= 0; n− − ) {    xC = ( xS << log2SbW ) + DiagScanOrder[ log2SbW ][ log2SbH ][ n ][ 0]     yC = ( yS << log2SbH ) + DiagScanOrder[ log2SbW ][ log2SbH ][ n ][1 ]     if( AbsLevel[ xC ][ yC ] > 0 )      TransCoeffLevel[ x0 ][ y0 ][cIdx ][ xC ][ yC ] =        ( 2 * AbsLevel[ xC ][ yC ] − ( QState > 1 ?1 : 0 ) ) *        ( 1 − 2 * coeff_sign_flag[ n ] )     QState =QStateTransTable[ QState ][ par_level_flag[ n ] ]   } else {   sumAbsLevel = 0    for( n = numSbCoeff − 1; n >= 0; n− − ) {     xC =( xS << log2SbW ) + DiagScanOrder[ log2SbW ][ log2SbH ][ n ][ 0 ]     yC= ( yS << log2SbH ) + DiagScanOrder[ log2SbW ][ log2SbH ][ n ][ 1 ]    if( AbsLevel[ xC ][ yC ] > 0 ) {      TransCoeffLevel[ x0 ][ y0 ][cIdx ][ xC ][ yC ] =        AbsLevel[ xC ][ yC ] * ( 1 − 2 *coeff_sign_flag[ n ] )      if( signHidden ) {       sumAbsLevel +=AbsLevel[ xC ][ yC ]       if( ( n = = firstSigScanPosSb ) && (sumAbsLevel % 2 ) = = 1 ) )        TransCoeffLevel[ x0 ][ y0 ][ cIdx ][xC ][ yC ] =          −TransCoeffLevel[ x0 ][ y0 ][ cIdx ][ xC ][ yC ]     }     }    }   }  } }2.5.2 Coefficients Coding of TS-Coded Blocks and QR-BDPCM Coded Blocks

QR-BDPCM follows the context modeling method for TS-coded blocks.

A modified transform coefficient level coding for the TS residual.Relative to the regular residual coding case, the residual coding for TSincludes the following changes:

-   -   (1) no signalling of the last x/y position    -   (2) coded_sub_block_flag coded for every subblock except for the        last subblock when all previous flags are equal to 0;    -   (3) sig_coeff_flag context modelling with reduced template,    -   (4) a single context model for abs_level gt1_flag and        par_level_flag,    -   (5) context modeling for the sign flag, additional greater than        5, 7, 9 flags,    -   (6) modified Rice parameter derivation for the remainder        binarization    -   (7) a limit for the number of context coded bins per sample, 2        bins per sample within one block.        2.5.2.1 Syntax and Semantics        7.3.6.10 Transform Unit Syntax

Descriptor transform_unit( x0, y0, tbWidth, tbHeight, treeType,subTuIndex ) { ...  if( tu_cbf_luma[ x0 ][ y0 ] && treeType !=DUAL_TREE_CHROMA    && ( tbWidth <= 32 ) && ( tbHeight <= 32 )    && (IntraSubPartitionsSplit[ x0 ][ y0 ] = = ISP_NO_SPLIT ) && ( !cu_sbt_flag) ) {   if( transform_skip_enabled_flag && tbWidth <= MaxTsSize &&tbHeight <= MaxTsSize )    transform_skip_flag[ x0 ][ y0 ] ae(v)   if((( CuPredMode[ x0 ][ y0 ] != MODE_INTRA &&sps_explicit_mts_inter_enabled_flag )    || ( CuPredMode[ x0 ][ y0 ] = =MODE_INTRA && sps_explicit_mts_intra_enabled_flag ))    && ( tbWidth <=32 ) && ( tbHeight <= 32 ) && ( !transform_skip_flag[ x0 ][ y0 ] ) )   tu_mts_idx[ x0 ][ y0 ] ae(v)  }  if( tu_cbf_luma[ x0 ][ y0 ] ) {  if( !transform_skip_flag[ x0 ][ y0 ] )    residual_coding( x0, y0,Log2( tbWidth ), Log2( tbHeight ), 0 )   else    residual_coding_ts( x0,y0, Log2( tbWidth ), Log2( tbHeight ), 0 )  }  if( tu_cbf_cb[ x0 ][ y0 ])   residual_coding( xC, yC, Log2( wC ), Log2( hC ), 1 )  if( tu_cbf_cr[x0 ][ y0 ] )   residual_coding( xC, yC, Log2( wC ), Log2( hC ), 2 ) }

Descriptor residual_ts_coding( x0, y0, log2TbWidth, log2TbHeight, cIdx ){  log2SbSize = ( Min( log2TbWidth, log2TbHeight) < 2 ? 1 : 2 ) numSbCoeff = 1 << ( log2SbSize << 1 )  lastSubBlock = ( 1 << (log2TbWidth + log2TbHeight − 2 * log2SbSize ) ) − 1 /* Loop oversubblocks from top-left (DC) subblock to the last one */  inferSbCbf = 1 MaxCcbs = 2 * ( 1 << log2TbWidth ) * ( 1<< log2TbHeight )  for( i =0; i<= lastSubBlock; i++ ) {   xS = DiagScanOrder[ log2TbWidth − log2SbSize][ log2TbHeight − log2SbSize ][ i ][ 0 ]   yS = DiagScanOrder[log2TbWidth − log2SbSize ][ log2TbHeight − log2SbSize ][ i ][ 1 ]   if(( i != lastSubBlock || !inferSbCbf )    coded_sub_block_flag[ xS ][ yS ]ae(v)    MaxCcbs− −   if( coded_sub_block_flag[ xS ][ yS ] && i <lastSubBlock )    inferSbCbf = 0   }  /* First scan pass */  inferSbSigCoeffFlag = 1   for( n = ( i = = 0; n <= numSbCoeff − 1; n++) {    xC = ( xS << log2SbSize ) + DiagScanOrder[ log2SbSize ][log2SbSize ][ n ][ 0 ]    yC = ( yS << log2SbSize ) + DiagScanOrder[log2SbSize ][ log2SbSize ][ n ][ 1 ]    if( coded_sub_block_flag[ xS ][yS ] &&     ( n == numSbCoeff − 1 || !inferSbSigCoeffFlag ) ) {    sig_coeff_flag[ xC ][ yC ] ae(v)     MaxCcbs− −     if(sig_coeff_flag[ xC ][ yC ] )      inferSbSigCoeffFlag = 0    }    if(sig_coeff_flag[ xC ][ yC ] ) {     coeff_sign_flag[ n ] ae(v)    abs_level_gtx_flag[ n ][ 0 ] ae(v)     MaxCcbs = MaxCcbs − 2     if(abs_level_gtx_flag[ n ][ 0 ] ) {      par_level_flag[ n ] ae(v)     MaxCcbs− −     }    }    AbsLevelPassX[ xC ][ yC ] =     sig_coeff_flag[ xC ][ yC ] + par_level_flag[ n ] +abs_level_gtx_flag[ n ][ 0 ]   }  /* Greater than X scan passes(numGtXFlags=5) */   for( i = 1; i <= 5 − 1 && abs_level_gtx_flag[ n ][i − 1 ]; i++ ) {    for( n = 0; n <= numSbCoeff − 1; n++ ) {     xC = (xS << log2SbSize ) + DiagScanOrder[ log2SbSize ][ log2SbSize ][ n ][ 0 ]    yC = ( yS << log2SbSize ) + DiagScanOrder[ log2SbSize ][ log2SbSize][ n ][ 1 ]     abs_level_gtx_flag[ n ][ i ] ae(v)     MaxCcbs− −    AbsLevelPassX[ xC ][ yC ] + = 2 * abs_level_gtx_flag[ n ][ i ]    }  }  /* remainder scan pass */   for( n = 0; n <= numSbCoeff − 1; n++ ){    xC = ( xS << log2SbSize ) + DiagScanOrder[ log2SbSize ][ log2SbSize][ n ][ 0 ]    yC = ( yS << log2SbSize ) + DiagScanOrder[ log2SbSize ][log2SbSize ][ n ][ 1 ]    if( abs_level_gtx_flag[ n ][ numGtXFlags − 1 ])     abs_remainder[ n ] ae(v)    TransCoeffLevel[ x0 ][ y0 ][ cIdx ][xC ][ yC ] = ( 1 − 2 * coeff_sign_flag[ n ] ) *             (AbsLevelPassX[ xC ][ yC ] + abs_remainder[ n ] )   }  } }

The number of context coded bins is restricted to be no larger than 2bins per sample for each CG.

TABLE 9-15 Assignment of ctxInc to syntax elements with context codedbins Syntax binIdx element 0 1 2 3 4 >=5 last_sig_coeff_x_prefix 0 . . .23 (clause 9.5.4.2.4) last_sig_coeff_y_prefix 0 . . . 23 (clause9.5.4.2.4) last_sig_coeff_x_suffix bypass bypass bypass bypass bypassbypass last_sig_coeff_y_suffix bypass bypass bypass bypass bypass bypasscoded_sub_block_flag[ ][ ] ( MaxCcbs > 0) ? (0..7 na na na na na (clause9.5.4.2.6) ) : bypass sig_coeff_flag[ ][ ] ( MaxCcbs > 0) ? (0..93 na nana na na (clause 9.5.4.2.8) ) : bypass par_level_flag[ ] ( MaxCcbs > 0)? (0..33 na na na na na (clause 9.5.4.2.9) ) : bypassabs_level_gtx_flag[ ][ i ] 0 . . . 70 na na na na na (clause 9.5.4.2.9)abs_remainder[ ] bypass bypass bypass bypass bypass bypassdec_abs_level[ ] bypass bypass bypass bypass bypass bypasscoeff_sign_flag[ ] bypass na na na na na transform_skip_flag[ x0 ][ y0 ]= = 0 coeff_sign_flag[ ] 0 na na na na na transform_skip_flag[ x0 ][ y0] = = 12.5 Chroma Direct Mode

In Direct Mode (DM), prediction mode of co-located luma block is usedfor deriving the chroma intra prediction mode.

Firstly, an intra prediction mode lumaIntraPredMode is derived:

-   -   If the co-located luma block is coded in MIP mode,        lumaIntraPredMode is set equal to Planar mode.    -   Otherwise, if the co-located luma block is coded in IBC mode or        palette mode, lumaIntraPredMode is set equal to DC mode.    -   Otherwise, lumaIntraPredMode is set equal to the intra        prediction mode of the co-located luma block covering the        corresponding luma sample of the center of chroma block. An        example is depicted in FIG. 25 .

Secondly, the intra chroma prediction mode (denoted as IntraPredModeC)is derived according to lumaIntraPredMode as highlighted in bold andItalic in the following table. Note that intra_chroma_pred_mode equal to4 refers to the DM mode.

TABLE 8-2 Specification of IntraPredModeC[ xCb ][ yCb ] depending oncclm_mode_flag, cclm_mode_idx, intra_chroma_pred_mode andlumaIntraPredMode lumaIntraPredMode X (0 <= cclm_mode_flag cclm_mode_idxintra_chroma_pred_mode 0 50 18 1 X <= 66) 0 — 0 66 0 0 0 0 0 — 1 50 6650 50 50 0 — 2 18 18 66 18 18 0 — 3 1 1 1 66 1 0 — 4 0 50 18 1 X 1 0 —81 81 81 81 81 1 1 — 82 82 82 82 82 1 2 — 83 83 83 83 83

Finally, if the color format of the picture is 4:2:2, IntraPredModeC isfurther modified according to the following table for the DM mode.

Specification of the 4:2:2 Mapping Process from Chromaintra PredictionMode X to Mode Y when Chroma_Format_Idc is Equal to 2

mode X 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 mode Y 0 1 61 62 6364 65 66 2 3 4 6 8 10 12 13 14 16 mode X 18 19 20 21 22 23 24 25 26 2728 29 30 31 32 33 34 35 mode Y 18 20 22 23 24 26 28 30 32 33 34 35 36 3738 39 40 41 mode X 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53mode Y 42 43 44 44 44 45 46 46 46 47 48 48 48 49 50 51 52 52 mode X 5455 56 57 58 59 60 61 62 63 64 65 66 mode Y 52 53 54 54 54 55 56 56 56 5758 59 60

The detailed draft is specified as follows.

8.43 Derivation Process for Chroma Intra Prediction Mode

Input to this process are:

-   -   a luma location (xCb, yCb) specifying the top-left sample of the        current chroma coding block relative to the top-left luma sample        of the current picture,    -   a variable cbWidth specifying the width of the current coding        block in luma samples,    -   a variable cbHeight specifying the height of the current coding        block in luma samples.        In this process, the chromaintra prediction m ode        IntraPredModeC[xCb] [yCb] is derived.        The corresponding luma intra prediction m ode lumaIntraPredMode        is derived as follows:    -   If intra_mip_flag[xCb] [yCb] is equal to 1, lumaIntraPredMode is        set equal to INTRA_PLANAR.    -   Otherwise, if CuPredMode[0] [xCb] [yCb] is equal to MODE_IBC or        MODE_PLT, lumaIntraPredMode is set equal to INTRA_DC.    -   Otherwise, lumaIntraPredMode is set equal to        IntraPredModeY[xCb+cbWidth/2] [yCb+cbHeight/2].        The chroma intra prediction mode IntraPredModeC[xCb] [yCb] is        derived using cclm_mode_mode_flag, cclm_mode_idx,        intra_chroma_pred_mode and lumaIntraPredMode as specified in        Table 8-2.

TABLE 8-2 Specification of IntraPredModeC[ xCb ][ yCb ] depending oncclm_mode_flag, cclm_mode_idx, intra_chroma_pred_mode andlumaIntraPredMode lumaIntraPredMode X (0 <= cclm_mode_flag cclm_mode_idxintra_chroma_pred_mode 0 50 18 1 X <= 66) 0 — 0 66 0 0 0 0 0 — 1 50 6650 50 50 0 — 2 18 18 66 18 18 0 — 3 1 1 1 66 1 0 — 4 0 50 18 1 X 1 0 —81 81 81 81 81 1 1 — 82 82 82 82 82 1 2 — 83 83 83 83 83When chroma_format_idc is equal to 2, the chroma intra prediction mode Yis derived using the chroma intra prediction mode X in Table 8-2 asspecified in Table 8-3, and the chroma intra prediction mode X is setequal to the chroma intra prediction mode Y afterwards.

TABLE 8-3 Specification of the 4:2:2 mapping process from chroma intraprediction mode X to mode Y when chroma_format_idc is equal to 2 mode X0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 mode Y 0 1 61 62 63 64 65 662 3 4 6 8 10 12 13 14 16 mode X 18 19 20 21 22 23 24 25 26 27 28 29 3031 32 33 34 35 mode Y 18 20 22 23 24 26 28 30 32 33 34 35 36 37 38 39 4041 mode X 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 mode Y42 43 44 44 44 45 46 46 46 47 48 48 48 49 50 51 52 52 mode X 54 55 56 5758 59 60 61 62 63 64 65 66 mode Y 52 53 54 54 54 55 56 56 56 57 58 59 60

3 Drawbacks of Existing Implementations

The current design has the following problems:

-   -   (1) The four pre-defined transform sets for chroma components is        the same as that for luma component. In addition, luma and        chroma blocks with the same intra prediction mode use the same        transform set. However, the chroma signal is typically smoother        compared to the luma component. Using the same set may be        sub-optimal.    -   (2) RST is only applied to certain CGs instead of all CGs.        However, the decision on signaling RST index is dependent on the        number of non-zero coefficients in the whole block. When all        coefficients in the RST-applied CGs are zeros, there is no need        to signal the RST index. However, the current design may still        signal the index which wastes unnecessary bits.    -   (3) RST index is signaled after residual coding since it        requires to record how many non-zero coefficients, whether there        exists non-zero coefficient in certain locations (e.g.,        numZeroOutSigCoeff, numSigCoeff in section 2.3.2.2.7). Such        design makes the parsing process more complex.    -   (4) RST index is context coded and context modeling is dependent        on the coded luma/chroma intra prediction mode, and MTS index.        Such design introduces parsing delay in terms of reconstruction        of intra prediction modes. And 8 contexts are introduced which        may be a burden for hardware implementation.        -   (a) DM and CCLM share the same context index offset which            doesn't make sense since they are two different chroma intra            prediction methods.    -   (5) The current design of non-TS residual coding firstly codes        the coefficients information, followed by the indices of RST        (i.e., use RST or not, if used, which matrix is selected). With        such design, the information of RST on/off couldn't be taken        into consideration in the entropy coding of residuals.    -   (6) RST is always applied to the top-left region of a transform        block with primary transform applied. However, for different        primary transform basis, it is not always true that the energy        is concentrated in the top-left region of a transform block.    -   (7 The determination of whether to signal RST related        information are conducted in different ways for dual-tree and        single-tree coding structure.    -   (8) When there are more than one TU in a CU (e.g. the CU size is        128×128), whether to parse the RST related information can only        be determined after decoding all the TUs. For example, for a        128×128 CU, the first PB could not be processed without waiting        for the LFNST index that comes after the last PB. Although this        does not necessarily break the overall 64×64 based decoder        pipeline (if the CABAC could be decoupled), it increases the        data buffering by 4× fora certain number of decoder pipeline        stages. It is costly.    -   (9) In JVET-00219, intra prediction mode of co-located luma        block is used for determining the LFNST transform set of CCLM        mode coded chroma blocks. However, the co-located luma block may        be coded with non-intra prediction modes (i.e., not the        conventional intra prediction method such as using        DC/Planar/Angular prediction direction). For example, in        dual-tree case, the co-located luma block may be coded in        palette mode, IBC mode etc. In this case, the intra prediction        mode of the co-located luma block is undefined. Therefore, how        to derive the secondary transform set for a chroma block is        unknown.    -   (10) The derivation of chroma intra prediction block defined in        sub-clause 8.4.4 checks whether the luma block covering the        corresponding top-left luma sample is coded with MIP/IBC/Palette        mode. If it is true, a default mode is given. Otherwise, the        intra prediction mode of the center luma sample is used. It will        cause two issues:        -   a. When the coding block covering the top-left luma sample            (e.g., TL in FIGS. 24 (a) and 24 (b), the corresponding luma            sample of the top-left chroma sample in the current chroma            block) is coded with MIP mode, and the coding block covering            the center luma sample (e.g., CR in FIG. 25 ) is coded with            normal intra mode, in this case, a default mode is used and            set to the chroma intra prediction mode which breaks the            correlation between current chroma block and the luma block            covering the CR. Lower coding performance will be caused.        -   b. When the coding block covering the top-left luma sample            (e.g., TL in FIGS. 24 (a) and 24 (b), the corresponding luma            sample of the top-left chroma sample in the current chroma            block) is coded with intra mode, and the coding block            covering the center luma sample (e.g., CR in FIG. 25 ) is            coded with IBC/Palette/MIP, in this case, the intra            prediction mode from the coding block covering CR is used to            derive the chroma DM mode. However, there is no definition            of such intra prediction modes associated with            MIP/IBC/Palette mode.    -   (11) In the last VVC working draft (which could be downloaded        from http://phenix.it-sudparis.eu/jvet/doc end        user/documents/15_Gothenburg/wg11/JVET-O2001-v14. zip), the        zero-out region size (i.e., nonZeroSize) is non-uniform for        RST4×4 and RST8×8. RST4×4 is applied when the block size is 4×N        or N×4 (N>=4). When N is 4 (i.e. the block size is 4×4), the        zero-out region size is equal to 8. When N is larger than 4,        nonZero Size is set equal to 16. For RST8×8, when the block size        is 8×8, the zero-out region size is equal to 8. Otherwise (i.e.,        the block width is larger than 8 and the block height is no        smaller than 8, or the block height is larger than 8 and the        block width is no smaller than 8), the zero-out region size is        equal to 16. The larger value of zero-out range brings higher        computational complexity at both encoder and decoder side.    -   (12) The memory usage for storing RST matrices is high.

4 Example Methods for Context Modeling for Residual Coding

Embodiments of the presently disclosed technology overcome the drawbacksof existing implementations, thereby providing video coding with highercoding efficiencies. The methods for context modeling for residualcoding, based on the disclosed technology, may enhance both existing andfuture video coding standards, is elucidated in the following examplesdescribed for various implementations. The examples of the disclosedtechnology provided below explain general concepts, and are not meant tobe interpreted as limiting. In an example, unless explicitly indicatedto the contrary, the various features described in these examples may becombined.

In the following description, a “block” may refer to coding unit (CU) ora transform unit (TU) or any rectangle region of video data. a “currentblock” may refer to a current being decoded/coded coding unit (CU) or acurrent being decoded/coded transform unit (TU) or any beingdecoded/coded coding rectangle region of video data. “CU” or “TU” may bealso known as “coding block” and” transform block”.

In these examples, the RST may be any variation of the design inJVET-N0193. RST could be any technology that may apply a secondarytransform to one block or apply a transform to the transform skip(TS)-coded block (e.g., the RST proposed in JVET-N0193 applied to theTS-coded block).

Hereinafter, “normal intra prediction mode” is used to refer to theconventional intra prediction method wherein the prediction signal isgenerated by extrapolating neighbouring pixels from a certain direction.such as DC mode, Planar mode and Angular intra prediction modes (e.g.,may further include wide angle intra prediction modes). Fora block codedwithout using normal intra prediction mode, the block may be coded withat least one of the coding methods, e.g., IBC, MIP, palette, BDPCMintra-prediction modes.

In addition, the ‘zero-out region’ or ‘zero-out CG’ may indicate thoseregions/CGs which always have zero coefficients due the reducedtransform size used in the secondary transform process. For example, ifthe secondary transform size is 16×32, and CG size is 4×4, it will beapplied to the first two CGs, but only the first CG may have non-zerocoefficients, the second 4×4 CG is also called zero-out CG.

Selection of Transform Matrices in RST

-   -   1. The sub-region that RST is applied to may be a sub-region        which is not the top-left part of a block.        -   a. In one example, RST may be applied to the top-right or            bottom-right or bottom-left or center sub-region of a block.        -   b. Which sub-region that RST is applied to may depend on the            intra prediction mode and/or primary transform matrix (e.g.,            DCT-II, DST-VII, Identity transform).    -   2. Selection of transform set and/or transform matrix used in        RST may depend on the color component.        -   a. In one example, one set of transform matrix may be used            for luma (or G) component, and one set for chroma components            (or B/R).        -   b. In one example, each color component may correspond to            one set.        -   c. In one example, at least one matrix is different in any            of the two or multiple sets for different color components.    -   3. Selection of transform set and/or transform matrix used in        RST may depend on intra prediction method (e.g., CCLM, multiple        reference line based intra prediction method, matrix-based intra        prediction method).        -   a. In one example, one set of transform matrix may be used            for CCLM coded blocks, and the other for non-CCLM coded            blocks.        -   b. In one example, one set of transform matrix may be used            for normal intra prediction coded blocks, and the other for            multiple reference line enabled blocks (i.e., which doesn't            use the adjacent line for intra prediction).        -   c. In one example, one set of transform matrix may be used            for blocks with joint chroma residual coding, and the other            for blocks which joint chroma residual coding is not            applied.        -   d. In one example, at least one matrix is different in any            of the two or multiple sets for different intra prediction            methods.        -   e. Alternatively, RST may be disabled for blocks coded with            certain intra prediction directions and/or certain coding            tools, e.g., CCLM, and/or joint chroma residual coding,            and/or certain color component (e.g., chroma).    -   4. A default intra prediction mode may be assigned for blocks        (e.g., CU/PU/CB/PB) which are not coded with normal intra        prediction mode, such as MIP, IBC, Palette.        -   a. The default mode may be dependent on the coding method            (MIP/IBC/Palette).        -   b. The default mode may be signaled or derived on-the-fly.        -   c. The default mode may be utilized in the derivation of            chroma derived mode (DM).        -   d. The default mode may be used to predict the            intra-prediction modes of other blocks. For example, it may            be utilized in the derivation of most-probable-mode (MPM)            list for coding subsequent blocks of current block.        -   e. The default mode assigned to a block of one color            component (e.g., luma) may be utilized in the derivation of            transform set or transform index of another color component            (e.g., chroma).        -   f. Alternatively, furthermore, the default modes may be            stored together with the prediction modes (e.g.,            intra/inter/IBC).        -   g. The default mode may NOT be assigned to inter coded            blocks.    -   5. It is proposed to use the information of one luma coding        block (which may also be known as the corresponding luma block)        covering the same pre-defined position of the chroma coding        block in all operations of the chroma intra prediction mode        derivation process. For example, the same luma coding block        covering the same pre-defined position of the chroma coding        block is used to check the prediction modes (or coding methods,        like IBC/MIP/Palette) of the luma coding block and used to fetch        the intra prediction mode of the luma coding block.        -   a. In one example, the same pre-defined position is defined            to be associated with the corresponding luma sample of the            center chroma sample of current chroma block (e.g., CR in            FIG. 25 ).        -   b. In one example, the same pre-defined position is defined            to be associated with the corresponding luma sample of the            top-left chroma sample of current chroma block (e.g., TL in            FIG. 24 b )).        -   c. Alternatively, furthermore, if the coding block covering            the same pre-defined position is coded with IBC/Palette/MIP            mode/any other non-normal intra-prediction mode, a default            intra prediction mode may be utilized to derive chroma intra            prediction mode. Otherwise, the decoded intra prediction            mode of the coding block covering the same pre-defined            position may be utilized to derive chroma intra prediction            mode.    -   6. For a chroma block coded with a non-normal intra prediction        mode (e.g., CCLM), when a corresponding luma block is coded with        non-normal intra prediction mode (e.g., IBC/Palette/MIP), the        transform set/transform matrix used in RST or other coding tools        may be derived from one or multiple default modes or default        transform set.        -   a. Whether to use default transform set/transform matrix or            derive the transform set/transform matrix from intra luma            prediction modes of the corresponding luma block may depend            on the coded mode of the corresponding luma block.            -   i. In one example, if the corresponding luma block is                coded with normal intra prediction mode and/or BDPCM,                the transform set/transform matrix may be derived                according to the luma intra prediction mode associated                with the corresponding luma block (e.g., co-located luma                block or the luma block defined in invention bullet 5).            -   ii. In one example, if the corresponding luma block is                coded with non-normal intra prediction mode (e.g.,                IBC/Palette and/or BDPCM), a default transform set                (e.g., with set index equal to K (e.g., K=0)) may be                used.        -   b. Whether to enable RST for blocks of one color component            may be dependent on the coding methods of one or multiple            corresponding blocks in another color component and/or the            coding method of current block.            -   i. Whether to enable RST for chroma blocks may be                dependent on the coding methods of one or multiple                corresponding luma blocks and/or the coding method of                current chroma block.            -   ii. In one example, if the corresponding luma block is                coded with normal intra prediction mode and/or BDPCM,                RST may be enabled for a chroma block.            -   iii. In one example, if the corresponding luma block is                coded with non-normal intra prediction mode (e.g.,                IBC/Palette and/or BDPCM), RST may be disabled for a                chroma block.                -   1) Alternatively, furthermore, if the corresponding                    luma block is coded with non-normal intra prediction                    mode (e.g., IBC/Palette and/or BDPCM), RST may be                    disabled for a chroma block coded with non-normal                    intra prediction mode (e.g., CCLM).        -   c. When the co-located luma block is not coded in a normal            intra prediction mode, a pre-defined intra prediction mode            may be assigned, which is then used for selection of the            transform set and/or transform matrix used in RST (e.g.,            according to the transform set selection table shown in            section 2.4.2.2.3).            -   i. In one example, if the co-located luma block is coded                in IBC (intra block copy) mode, a first pre-defined                intra prediction mode (e.g., DC or planar mode) may be                assigned.            -   ii. In one example, if the co-located luma block is                coded in palette mode, a second pre-defined intra                prediction mode (e.g., DC or planar mode) may be                assigned.            -   iii. In one example, if the co-located luma block is                coded in MIP mode, a third pre-defined intra prediction                mode (e.g., Planar or DC mode) may assigned.            -   iv. In one example, if the co-located luma block is                coded in BDPCM (Block-based Delta Pulse Code Modulation)                mode, a fourth pre-defined intra prediction mode (e.g.,                Planar or DC mode) may be assigned.            -   v. In one example, for each of the pre-defined intra                prediction modes, it may be DC mode or Planar mode or                Vertical mode or Horizontal mode or 45-degree Mode or                135-degree mode.            -   vi. Alternatively, if the co-located luma block is coded                in MIP mode, the MIP mode may be mapped to an intra                prediction mode according to the MIP mode and the                dimension of the co-located luma block, e.g., by using                the table “Specification of mapping between MIP and                intra prediction modes” in section 2.2.2.2.            -   vii. The predefined mode may be adaptively selected from                several candidates, such as DC mode and Planar mode.                -   1) The predefined mode may be signaled from the                    encoder to the decoder.                -   2) The predefined mode may be derived from the                    encoder to the decoder.                -    a. For example, the predefined mode may be DC mode                    if the decoder determines that the current picture                    is a screen content picture, otherwise, the                    predefined mode is the Planar mode.            -   viii. The predefined mode may be defined in the same way                as those utilized for the chroma DM derivation process                (e.g., lumaIntraPredMode in sub-clause 8.4.3).        -   d. When the co-located luma block is coded in a normal intra            prediction mode or/and BDPCM mode, a chroma intra prediction            mode may be derived dependent on the intra prediction mode            of the co-located luma block, in the same way as the DM mode            (such as specified in section 2.7). The derived chroma intra            prediction mode is then used for selection of the transform            set and/or transform matrix used in RST (e.g., according to            the transform set selection table shown in section 2.6).        -   e. In above bullets, the co-located luma block may be the            coding block covering a specific luma position, such as TL            in FIG. 24 b or CR in FIG. 25 .    -   7. Selection of transform set and/or transform matrices used in        RST may depend on the primary transform.        -   a. In one example, the primary transform may include but not            limited to variances of DCT-II, DST-VII, DCT-VIII, Transform            skip mode, Identity transform, or transform from training            (like KLT based transform.        -   b. In one example, if the primary transform applied to one            block is the identity transform (e.g., TS mode is applied to            one block), the transform set and/or transform matrices used            in RST may be different from other kinds of primary            transform.            -   i. In one example, one set of transform matrices may be                used to transform skip mode.                -   1) Alternatively, furthermore, one syntax element                    may be signaled to indicate whether RST is used or                    not for transform skip mode or which matrix among                    the set of transform matrices is used for a block.        -   c. In one example, if the horizontal and vertical 1-D            primary transform applied to one block is the same basis            (e.g., both DCT-II), the transform set and/or transform            matrices used in RST may be different from that primary            transforms from different basis for different directions            (vertical or horizontal).        -   d. In one example, one set of transform matrices may be used            for DCT-II (or the first primary transform), and one or            multiple sets of transform matrices may be used for other            primary transform modes.        -   e. In one example, each primary transform mode may            correspond to one RST matrix set.        -   f. In one example, at least one matrix in one set is            different in any of the two or multiple sets for different            primary transform modes.    -   8. Selection of transform set and/or transform matrices used in        RST may depend on block dimension (width/height, or ratio)/block        shape (square or non-square).        -   a. In one example, multiple sets of transform matrices may            be used, and each set may be corresponding to one category            of blocks wherein the category is determined by block            dimension/block shape.    -   9. Multiple sets of transform matrices may be enabled for        encoding/decoding a video processing unit (e.g.,        sequence/video/picture/slice/tile/brick/subpicture), and        indication of which set(s) among those multiple sets may be        utilized may be signaled in picture/slice/tile/brick/subpicture        level.        Signaling of RST Side Information and Residual Coding    -   10. Whether to and/how to signal the side information of RST        (e.g., st_idx) may depend on the last non-zero coefficient (in        scanning order) in the block.        -   a. In one example, only if the last non-zero coefficient is            located in the CGs that RST applied to, RST may be enabled,            and the index of RST may be signaled.        -   b. In one example, if the last non-zero coefficient is not            located in the CGs that RST applied to, RST is disabled and            signaling of RST is skipped.    -   11. Whether to and/how to signal the side information of RST        (e.g., st_idx) may depend on coefficients of certain color        component instead of all available color components in a CU.        -   a. In one example, only the luma information may be utilized            to determine whether to and/how to signal the side            information of RST.            -   i. Alternatively, furthermore, the above method is                applied only when a block's dimension satisfied certain                conditions.                -   1) The conditions are W<T1 or H<T2.                -   2) For example, T1=T2=4. Therefore, for 4×4 CU, the                    luma block size is 4×4, two chroma blocks in 4:2:0                    format is 2×2, in this case, only luma information                    may be utilized.            -   ii. Alternatively, furthermore, the above method is                applied only when current partition type tree is single                tree.        -   b. Whether to use one color component's information or all            color components' information may depend on the block            dimension/coded information.    -   12. Whether to and/how to signal the side information of RST        (e.g., st_idx) may depend on coefficients within a partial        region of one block instead of the whole block.        -   a. In one example, partial region may be defined as the CGs            that RST is applied to.        -   b. In one example, partial region may be defined as the            first or last M (e.g., M=1, or 2) CGs in scanning order or            reverse scanning order of the block.            -   i. In one example, M may depend on block dimension.            -   ii. In one example, M is set to 2 if block size is 4×N                and/or N×4 (N>8).            -   iii. In one example, M is set to 1 if block size is 4×8                and/or 8×4 and/or W×H(W>=8, H>=8).        -   c. In one example, information of a block (e.g., the number            of non-zero coefficients of a block) with dimensions W×H may            be disallowed to be taken into consideration to determine            the usage of RST and/or signaling of RST related            information.            -   i. For example, the number of non-zero coefficients of a                block may not be counted if W<T1 or H<T2. For example,                T1=T2=4.        -   d. In one example, the partial region may be defined as the            top-left M×N region of the current block with dimensions            W×H.            -   i. In one example, M may be smaller than W and/or N may                be smaller than H.            -   ii. In one example, M and N may be fixed numbers. E.g.                M=N=4.            -   iii. In one example, M and/or N may depend on W and/or                H.            -   iv. In one example, M and/or N may depend on the maximum                allowed transform size.                -   1) For example, M=8 and N=4 if W is greater than 8                    and H is equal to 4.                -   2) For example, M=4 and N=8 if H is greater than 8                    and W is equal to 4.                -   3) For example, M=4 and N=4 if the none of the above                    two conditions is satisfied.            -   v. Alternatively, furthermore, these methods may be                applied only for certain block dimensions, such as the                conditions in 7.c is not satisfied.        -   e. In one example, the partial region may be the same to all            blocks.            -   i. Alternatively, it may be changed based on the block                dimension, and/or coded information.        -   f. In one example, the partial region may depend on the            given range of scanning order index.            -   i. In one example, the partial region may be that                covering coefficients located in a specific range with                their scanning order index within [dxS, IdxE],                inclusively, based on the coefficient scanning order                (e.g., the inversed decoding order) of the current block                with dimensions W×H.                -   1) In one example, IdxS is equal to 0.                -   2) In one example, IdxE may be smaller than W×H−1.                -   3) In one example, IdxE may be fixed numbers. E.g.                    IdxE=15.                -   4) In one example, IdxE may depend on W and/or H.        -   a. For example, IdxE=31 if W is greater than 8 and H is            equal to 4.        -   b. For example, IdxE=31 if H is greater than 8 and W is            equal to 4.        -   c. For example, IdxE=7 if W is equal to 8 and H is equal to            8.        -   d. For example, IdxE=7 if W is equal to 4 and H is equal to            4.        -   e. For example, IdxE=15 if the none of the above two            conditions a) and b) is satisfied.        -   f. For example, IdxE=15 if the none of the above two            conditions a), b), c) and d) is satisfied.        -   g. For example, IdxE=15 if the none of the above two            conditions c) and d) is satisfied.            -   ii. Alternatively, furthermore, these methods may be                applied only for certain block dimensions, such as the                conditions in 7.c is not satisfied.        -   g. In one example, it may depend on the position of non-zero            coefficients within a partial region.        -   h. In one example, it may depend on the energy (such as sum            of squares or sum of absolute values) of non-zero            coefficients within a partial region.        -   i. In one example, it may depend on the number of non-zero            coefficients within a partial region of one block instead of            the whole block.            -   i. Alternatively, it may depend on the number of                non-zero coefficients within a partial region of one or                multiple blocks in the CU.            -   ii. When the number of non-zero coefficients within                partial region of one block is less than a threshold,                signaling of the side information of RST may be skipped.            -   iii. In one example, the threshold is fixed to be N                (e.g., N=1 or 2).            -   iv. In one example, the threshold may depend on the                slice type/picture type/partition tree type (dual or                single)/video content (screen content or camera captured                content).            -   v. In one example, the threshold may depend on color                formats such as 4:2:0 or 4:4:4, and/or color components                such as Y or Cb/Cr.    -   13. When there are no non-zero coefficients in the CGs that RST        may be applied to, RST shall be disabled.        -   a. In one example, when RST is applied to one block, at            least one CG that RST is applied to must contain at least            one non-zero coefficient.        -   b. In one example, for 4×N and/or N×4 (N>8), if RST is            applied, the first two 4×4 CGs must contain at least one            non-zero coefficient.        -   c. In one example, for 4×8 and/or 8×4, if RST is applied,            the top-left 4×4 must contain at least one non-zero            coefficient.        -   d. In one example, for W×H(W>=8 and H>=8), if RST is            applied, the top-left 4×4 must contain at least one non-zero            coefficient.        -   e. A conformance bitstream must satisfy one or multiple of            above conditions.    -   14. RST related syntax elements may be signaled before coding        residuals (e.g., transform coefficients/directly quantized).        -   a. In one example, the counting of number of non-zero            coefficients in the Zero-out region (e.g.,            numZeroOutSigCoeff) and number of non-zero coefficients in            the whole block (e.g., numSigCoeff) is removed in the            parsing process of coefficients.        -   b. In one example, the RST related syntax elements (e.g,            st_idx) may be coded before residual_coding.        -   c. RST related syntax elements may be conditionally signaled            (e.g., according to coded block flags, TS mode usage).            -   vi. In one example, the RST related syntax elements                (e.g, st_idx) may be coded after the signaling of coded                block flags or after the signaling of TS/MTS related                syntax elements.            -   vii. In one example, when TS mode is enabled (e.g., the                decoded transform_skip_flag is equal to 1), the                signaling of RST related syntax elements is skipped.        -   d. Residual related syntax may not be signaled for zero-out            CGs.        -   e. How to code residuals (e.g., scanning order,            binarization, syntax to be decoded, context modeling) may            depend on the RST.            -   i. In one example, raster scanning order instead of                diagonal up-right scanning order may be applied.                -   1) The raster scanning order is from left to right                    and top to below, or in the reverse order.                -   2) Alternatively, vertical scanning order (from top                    to below and from left to right, or in the reverse                    order) instead of diagonal up-right scanning order                    may be applied.                -   3) Alternatively, furthermore, context modeling may                    be modified.                -    a. In one example, the context modeling may depend                    on the previously coded information in a template                    which are the most recent N neighbors in the scan                    order, instead of using right, bottom, bottom-right                    neighbors.                -    b. In one example, the context modeling may depend                    on the previously coded information in a template                    according to the scanned index (e.g., −1, −2, . . .                    assuming current index equal to 0).            -   ii. In one example, different binarization methods                (e.g., rice parameter derivation) may be applied to code                the residuals associated with RST-coded and                non-RST-coded blocks.            -   iii. In one example, signaling of certain syntax                elements may be skipped for RST coded blocks.                -   1) Signaling of the CG coded block flags                    (coded_sub_block_flag) for the CGs that RST is                    applied to may be skipped.                -    a. In one example, when RST8×8 applied to the first                    three CGs in diagonal scan order, signaling of CG                    coded block flags is skipped for the second and                    third CGs, e.g., the top-right 4×4 CG and left-below                    4×4 CG in the top-left 8×8 region of the block.                -    i. Alternatively, furthermore, the corresponding CG                    coded block flag is inferred to be 0, i.e., all                    coefficients are zero.                -    b. In one example, when RST is applied to one                    block, signaling of CG coded block flag is skipped                    for the first CG in the scanning order (or the last                    CG in the reverse scanning order).                -    ii. Alternatively, furthermore, the CG coded block                    flag for the top-left CG in the block is inferred to                    be 1, i.e., it contains at least one non-zero                    coefficient.                -    c. An example of 8×8 block is depicted in FIG. 21 .                    When RST8×8 or RST4×4 is applied to the 8×8 block,                    coded_sub_block_flag of CG0 is inferred to be 1,                    coded_sub_block_flag of CG1 and CG2 are inferred to                    be 0.                -   2) Signaling of the magnitudes of coefficients                    and/or the sign flags for certain coordinates may be                    skipped.                -    a. In one example, if the index relative to one CG                    in a scan order is no less than the maximum allowed                    index that non-zero coefficient may exist (e.g.,                    nonZero Size in section 0), the signaling of                    coefficients may be skipped.                -    b. In one example, signaling of the syntax                    elements, such as sig_coeff_flag,                    abs_level_gtX_flag, par_level_flag, abs_remainder,                    coeff_sign_flag, dec_abs_level may be skipped.                -   3) Alternatively, signaling of residuals (e.g., CG                    coded block flags, the magnitudes of coefficients                    and/or the sign flags for certain coordinates) may                    be kept, however, the context modeling may be                    modified to be different from other CGs.            -    iv. In one example, the coding of residuals in                RST-applied CGs and other CGs may be different.                -   1) For above sub-bullets, they may be applied only                    to the CGs which RST are applied.    -   15. RST related syntax elements may be signaled before other        transform indications, such as transform skip and/or MTS index.        -   a. In one example, the signaling of transform skip may            depend on RST information.            -   i. In one example, transform skip indication is not                signaled and inferred to be 0 for a block if RST is                applied in the block.        -   b. In one example, the signaling of MTS index may depend on            RST information.            -   i. In one example, one or multiple MTS transform                indication is not signaled and inferred to be not used                for a block if RST is applied in the block.    -   16. It is proposed to use different context modeling methods in        arithmetic coding for different parts within one block.        -   a. In one example, the block is treated to be two parts, the            first M CGs in the scanning order, and remaining CGs.            -   i. In one example, M is set to 1.            -   ii. In one example, M is set to 2 for 4×N and N×4 (N>8)                blocks; and set to 1 for all the other cases.        -   b. In one example, the block is treated to be two parts,            sub-regions where RST is applied, and sub-regions where RST            is not applied.            -   i. If RST4×4 is applied, the RST applied sub-region is                the first one or two CGs of the current block.            -   ii. If RST4×4 is applied, the RST applied sub-region is                the first three CGs of the current block.        -   c. In one example, it is proposed to disable the usage of            previously coded information in the context modeling process            for the first part within one block but enable it for the            second part.        -   d. In one example, when decoding the first CG, the            information of the remaining one or multiple CGs may be            disallowed to be used.            -   i. In one example, when coding the CG coded block flag                for the first CG, the value of the second CG (e.g.,                right or below) is not taken into consideration.            -   ii. In one example, when coding the CG coded block flag                for the first CG, the value of the second and third CG                (e.g., right and below CGs for W×H(W>=8 and H>=8)) is                not taken into consideration.            -   iii. In one example, when coding the current                coefficient, if its neighbor in the context template is                in a different CG, the information from this neighbor is                disallowed to be used.        -   e. In one example, when decoding coefficients in the RST            applied region, the information of the rest region that RST            is not applied to may be disallowed to be used.        -   f. Alternatively, furthermore, the above methods may be            applied under certain conditions.            -   i. The condition may include whether RST is enabled or                not.            -   ii. The condition may include the block dimension.                Context Modeling in Arithmetic Coding of RST Side                Information    -   17. When coding the RST index, the context modeling may depend        on whether explicit or implicit multiple transform selection        (MTS) is enabled.        -   a. In one example, when implicit MTS is enabled, different            contexts may be selected for blocks coded with same intra            prediction modes.            -   i. In one example, the block dimensions such as shape                (square or non-square) is used to select the context.        -   b. In one example, instead of checking the transform index            (e.g, tu_mts_idx) coded for the explicit MTS, the transform            matrix basis may be used instead.            -   i. In one example, for transform matrix basis with                DCT-II for both horizontal and vertical 1-D transforms,                the corresponding context may be different from other                kinds of transform matrices.    -   18. When coding the RST index, the context modeling may depend        on whether CCLM is enabled or not (e.g., sps_cclm_enabled_flag).        -   a. Alternatively, whether to enable or how to select the            context for RST index coding may depend on whether CCLM is            applied to one block.        -   b. In one example, the context modeling may depend on            whether CCLM is enabled for current block.            -   i. In one example, the                intraModeCtx=sps_cclm_enabled_flag?                (intra_chroma_pred_mode[x0] [y0] is CCLM:                intra_chroma_pred_mode[x0] [y0] is DM)? 1:0.        -   c. Alternatively, whether to enable or how to select the            context for RST index coding may depend on whether the            current chroma block is coded with the DM mode.            -   i. In one example, the                intraModeCtx=(intra_chroma_pred_mode[x0]                [y0]=(sps_cclm_enabled_flag? 7:4))? 1:0.    -   19. When coding the RST index, the context modeling may depend        on the block dimension/splitting depth (e.g., quadtree depth        and/or BT/TT depth).    -   20. When coding the RST index, the context modeling may depend        on the color formats and/or color components.    -   21. When coding the RST index, the context modeling may be        independent from the intra prediction modes, and/or the MTS        index.    -   22. When coding the RST index, the first and/or second bin may        be context coded with only one context; or bypass coded.        -   a. In one example, the first bin may be context coded and            context modeling is purely dependent on the partitioning            tree type (i.e., single tree or dual tree), and the second            bin is bypass coded.            -   i. Alternatively, furthermore, the above method is                invoked for a given slice type. For different slice                types, different contexts may be utilized.                Invoking RST Process Under Conditions    -   23. Whether to invoke the inverse RST process may depend on the        CG coded block flags.        -   a. In one example, if the top-left CG coded block flag is            zero, there is no need invoke the process.            -   i. In one example, if the top-left CG coded block flag                is zero and the block size is unequal to 4×N/N×4 (N>8),                there is no need invoke the process.        -   b. In one example, if the first two CG coded block flags in            the scanning order are both equal to zero, there is no need            invoke the process.            -   i. In one example, if the first two CG coded block flags                in the scanning order are both equal to zero and the                block size is equal to 4×N/N×4 (N>8), there is no need                invoke the process.    -   24. Whether to invoke the inverse RST process may depend on        block dimension.        -   a. In one example, for certain block dimensions, such as            4×8/8×4, RST may be disabled. Alternatively, furthermore,            signaling of RST related syntax elements may be skipped.            Unification for Dual-Tree and Single Tree Coding    -   25. The usage of RST and/or signaling of RST related information        may be determined in the same way in the dual-tree and single        tree coding.        -   a. For example, when the number of counted non-zero            coefficients (e.g numSigCoeff specified in JVET-N0193) is            not larger than T1 in the dual-tree coding case or not            larger than T2 in the single-tree coding, RST should not be            applied, and the related information is not signaled,            wherein T1 is equal to T2.        -   b. In one example, T1 and T2 are both set to N, e.g., N=1 or            2.            Considering Multiple TUs in a CU.    -   26. Whether to and/or how to apply RST may depend on the block        dimensions W×H.        -   a. In one example, when RST may not be applied if the W>T1            or H>T2.        -   b. In one example, when RST may not be applied if the W>T1            and H>T2.        -   c. In one example, when RST may not be applied if the            W*H>=T.        -   d. For above examples, the following apply:            -   i. In one example, the block is a CU.            -   ii. In one example, T1=T2=64.            -   iii. In one example, T1 and/or T2 may depend on the                allowed maximum transform size. E.g. T1=T2=the allowed                maximum transform size. iv. In one example, T is set to                4096.        -   e. Alternatively, furthermore, if RST is determined not to            be applied, related information may not be signaled.    -   27. When there are N(N>1) TUs in a CU, coded information of only        one of the N TUs is used to determine the usage of RST and/or        signaling of RST related information.        -   a. In one example, the first TU of the CU in decoding order            may be used to make the determination.        -   b. In one example, the top-left TU of the CU in decoding            order may be used to make the determination.        -   c. In one example, the determination with the specific TU            may be made in the same way to the case when there is only            one TU in the CU.    -   28. Usage of RST and/or signaling of RST related information may        be performed in the TU-level or PU-level instead of CU-level.        -   a. Alternatively, furthermore, different TUs/PUs within a CU            may choose different secondary transform matrices or            enabling/disabling control flags.        -   b. Alternatively, furthermore, for the dual tree case and            chroma blocks are coded, different color components may            choose different secondary transform matrices or            enabling/disabling control flags.        -   c. Alternatively, whether to signal RST related information            in which video unit level may depend on the partition tree            type (dual or single).        -   d. Alternatively, whether to signal RST related information            in which video unit level may depend on the relationship            between a CU/PU/TU and maximum allowed transform block            sizes, such as larger or smaller.            Unification of Zero-Out Range.    -   29. The zero-out region may depend on the dimensions (e.g. size)        of the sub-block that RST is applied to.        -   a. In one example, suppose nonZeroSize is equal to            nonZeroSizeA when the size of the current sub-block is M×N            and nonZeroSize (i.e., number of samples in the Zero-out            region) is equal to nonZeroSizeB when the size of the            current sub-block is K×L. When Min(M, N) is no smaller than            Min(K, L), then nonZeroSizeA is no smaller than            nonZeroSizeB. For example, M=8, N=8, K=4, L=4, and            nonZeroSizeA=16, nonZeroSizeB=8.        -   b. In one example, if the sub-block size is 4×4, (e.g.,            either width or height of a block is equal to 4), the            nonZeroSize may be set to a fixed value L (e.g., L is equal            to 8).            -   i. In one example, for 4×N or N×4 blocks, the                nonZeroSize may be set to a fixed value L.        -   c. In one example, if the sub-block size is 8×8, (e.g.,            either width or height of a block is equal to 8), the            nonZeroSize may be set to a fixed value L (e.g., L is equal            to 8).            -   i. In one example, for 8×N or N×8 blocks, the                nonZeroSize may be set to a fixed value L.        -   d. Alternatively, the nonZeroSize may be set to a fixed            value L (e.g., L is equal to 8) regardless of the block            sizes.    -   30. The zero-out region may depend on the secondary transform        size.        -   e. In one example, it may depend on whether RST4×4 or RST8×8            is applied.        -   f. In one example, suppose nonZeroSize (i.e., number of            samples in the Zero-out region) is equal to nonZeroSizeA            when RST8×8 is applied and nonZeroSize is equal to            nonZeroSizeB when RST4×4 is applied. Then nonZeroSizeA is no            smaller than nonZeroSizeB. For example, nonZeroSizeA=16 and            nonZeroSizeB=8.            Reduction of the Dimension of the Transform Matrices.    -   31. The dimension(s) of the transform and/or inverse-transform        matrices for RST may be reduced.        -   g. In one example, the dimensions of the transform matrices            of RST4×4 can be reduced to ½, ¼, or ⅛ of its original            dimension(s). For example, the original dimension(s) of the            transform matrices of RST4×4 is 16×16. The dimensions can be            reduced to 16×8, 16×4 or 16×2.        -   h. In one example, the dimension(s) of the transform and/or            inverse-transform matrices of RST8×8 can be reduced to ½, ¼,            or ⅛ of its original dimension(s). For example, the            dimension of the transform matrices of RST8×8 may be reduced            to 48×8, 48×4 or 48×2 from 48×16.            -   i. Alternatively, furthermore, the zero-out region may                be dependent on the dimension of reduced dimension.    -   32. One or multiple residual patterns may be defined. A residual        pattern may be a block with masks to define wherein the        coefficients after inverse transform may be assigned to. For        example, for a region with dimensions M*N, only partial of them        are allowed to have coefficients.        -   a. In one example, the pattern is defined in FIG. 26 ,            wherein the highlighted part may be associated with            transform coefficients and the remaining parts are            associated with zero coefficients.            -   i. In one example, the pattern may be not consecutive,                that is, one position may be associated with                coefficient, and four of its neighboring positions are                not associated with coefficient.        -   b. In one example, the above pattern may be applied to            certain blocks such as 8×8 and/or M×N (M>8, N>8) regardless            the signaled transform index.        -   c. In one example, whether and/or how to apply the above            pattern may depend on the which transform(s) is (are) used            in the current block (e.g., whether RST is applied, or            certain primary transform (e.g., DCT2) is applied).            -   i. For example, the pattern is applied only when RST is                applied.                -   1) For example, the pattern is applied when and only                    when RST is applied on a M×N (M>=8, N>=8) block                    regardless the signaled transform index.                -    a. Alternatively, furthermore, bullet 27-29 may be                    applied.            -   ii. For example, the pattern is applied when a certain                transform is used.        -   d. In one example, there may be multiple residual patterns            defined or derived from a larger residual pattern with more            coefficients allowed.            -   iii. In one example, for each residual pattern, multiple                transform matrices may be defined.                -   1) Alternatively, furthermore, the indication of                    selected residual pattern as well as the transform                    among the multiple ones associated with the selected                    pattern may be further signaled.            -   iv. In one example, a residual pattern may be derived by                excluding positions which are associated with smaller                transform coefficients in a K*L transform matrix.                -   1) In one example, a first residual pattern may be                    derived by excluding positions which are associated                    with the first K0 smallest transform coefficients.                    Therefore, (K*L−K0) positions may be assigned with                    residuals.                -   2) In one example, a second residual pattern may be                    derived by excluding positions which are associated                    with the first K1 smallest transform coefficients                    wherein K1 is unequal to K0. Therefore, (K*L−K1)                    positions may be assigned with residuals.        -   e. In one example, the above examples may be applied to            blocks smaller than maximum transform blocks.        -   f. In one example, how to select the pattern may depend on            color component/coded information of current block and/or            its neighboring blocks.        -   g. In one example, ‘RST is applied to a block’ means that            secondary transform is applied to that block (e.g., LFNST            index unequal to 0 (e.g., equal to 1 or 2)).            Retrained LFNST Matrices and Signaling    -   33. The RST matrices may be modified.        -   a. In one example, the 16×16 RST matrices may be modified to            be lowFreqTransMatrix shown in 0 with nTrS equal to 16.        -   b. In one example, the 16×48 RST matrices may be modified to            be lowFreqTransMatrix shown in 0 with nTrS equal to 48.    -   34. The RST matrices may be modified by online retraining.    -   a. In one example, the RST matrices may be retrained from the        reconstructed        -   transform coefficient blocks, and eigen decomposition (or            spectral decomposition) may be applied to calculate the            eigenvalues and eigenvectors. The eigenvectors or the sorted            eigenvectors may constitute the retrained RST matrices.            -   i. In one example, the RST retraining may be applied at                both the encoder and the decoder.            -   ii. In one example, the RST retraining may be only                applied at the encoder.        -   b. In one example, the RST matrices may be modified only            after encoding or decoding an intra slice (or intra            picture).        -   c. In one example, the retrained RST matrices may be used to            replace the original RST matrices.            -   i. Alternatively, the retrained RST matrices may be used                to replace part of the original RST matrices.        -   d. In one example, the retrained RST matrices may be used as            additional set of RST matrices.        -   e. In one example, the retrained RST matrices may be            signaled from the encoder to the decoder.            -   i. In one example, the retrained RST matrices may be                signaled in Slice header, Picture header (or Picture                Parameter Set (PPS)), or Adaption Parameter Set (APS).            -   ii. In one example, the difference values between the                original RST matrix and the retrained RST matrix may be                signaled.                -   1) Alternatively, the difference values between the                    RST matrix previously signaled and the retrained RST                    matrix may be signaled.            -   iii. In one example, only part of the retrained RST                matrix may be signaled.                -   1) In one example, only N sets of the retrained M                    sets of RST matrices (N<M) are signaled.                -   2) In one example, only top-left W*H sub-matrix in                    an RST matrix is signaled.

The examples described above may be incorporated in the context of themethods described below, e.g., methods 2200, 2210, 2220, 2230, 2240 and2250, which may be implemented at a video decoder or a video encoder.

FIG. 22A shows a flowchart of an exemplary method for video processing.The method 2200 includes, at step 2202, selecting, based on acharacteristic of a current video block, a transform set or a transformmatrix for an application of a reduced secondary transform to thecurrent video block.

The method 2200 includes, at step 2204, applying, as part of aconversion between the current video block and a bitstreamrepresentation of a video comprising the current video block, theselected transform set or transform matrix to a portion of the currentvideo block.

In some embodiments, the portion of the current video block is atop-right sub-region, bottom-right sub-region, bottom-left sub-region orcenter sub-region of the current video block.

In some embodiments, the characteristic of the current video block is anintra prediction mode or a primary transform matrix of the current videoblock.

In some embodiments, the characteristic is a color component of thecurrent video block. In an example, a first transform set is selectedfor a luma component of the current video block, and wherein a secondtransform set different from the first transform set is selected for oneor more chroma components of the current video block.

In some embodiments, the characteristic is an intra prediction mode oran intra coding method of the current video block. In an example, theintra prediction method comprises a multiple reference line (MRL)-basedprediction method or a matrix-based intra prediction method. In anotherexample, a first transform set is selected when the current video blockis a cross-component linear model (CCLM) coded block, and wherein asecond transform set different from the first transform set is selectedwhen the current video block is a non-CCLM coded block. In yet anotherexample, a first transform set is selected when the current video blockis coded with a joint chroma residual coding method, and wherein asecond transform set different from the first transform set is selectedwhen the current video block is not coded with the joint chroma residualcoding method.

In some embodiments, the characteristic is a primary transform of thecurrent video block.

FIG. 22B shows a flowchart of an exemplary method for video processing.The method 2210 includes, at step 2212, making a decision, based on oneor more coefficients associated with a current video block, regarding aselective inclusion of signaling of side information for an applicationof a reduced secondary transform (RST) in a bitstream representation ofthe current video block.

The method 2210 includes, at step 2214, performing, based on thedecision, a conversion between the current video block and a videocomprising the bitstream representation of the current video block.

In some embodiments, the one or more coefficients comprises a lastnon-zero coefficient in a scanning order of the current video block.

In some embodiments, the one or more coefficients comprises a pluralityof coefficients within a partial region of the current video block. Inan example, the partial region comprises one or more coding groups thatthe RST could be applied to. In another example, the partial regioncomprises a first M coding groups or a last M coding groups in ascanning order of the current video block. In yet another example, thepartial region comprises a first M coding groups or a last M codinggroups in a reverse scanning order of the current video block. In yetanother example, making the decision is further based on an energy ofone or more non-zero coefficients of the plurality of coefficients.

FIG. 22C shows a flowchart of an exemplary method for video processing.The method 2220 includes, at step 2222, configuring, for an applicationof a reduced secondary transform (RST) to a current video block, abitstream representation of the current video block, wherein a syntaxelement related to the RST is signaled in the bitstream representationbefore coding residual information.

The method 2220 includes, at step 2224, performing, based on theconfiguring, a conversion between the current video block and thebitstream representation of the current video block.

In some embodiments, signaling the syntax element related to the RST isbased on at least one coded block flag or a usage of a transformselection mode.

In some embodiments, the bitstream representation excludes the codingresidual information corresponding to coding groups with all zerocoefficients.

In some embodiments, the coding residual information is based on theapplication of the RST.

FIG. 22D shows a flowchart of an exemplary method for video processing.The method 2230 includes, at step 2232, configuring, for an applicationof a reduced secondary transform (RST) to a current video block, abitstream representation of the current video block, wherein a syntaxelement related to the RST is signaled in the bitstream representationbefore either a transform skip indication or a multiple transform set(MTS) index.

The method 2230 includes, at step 2234, performing, based on theconfiguring, a conversion between the current video block and thebitstream representation of the current video block.

In some embodiments, the transform skip indication or the MTS index isbased on the syntax element related to the RST.

FIG. 22E shows a flowchart of an exemplary method for video processing.The method 2240 includes, at step 2242, configuring, based on acharacteristic of a current video block, a context model for coding anindex of a reduced secondary transform (RST).

The method 2240 includes, at step 2244, performing, based on theconfiguring, a conversion between the current video block and abitstream representation of a video comprising the current video block.

In some embodiments, the characteristic is an explicit or implicitenablement of a multiple transform selection (MTS) process.

In some embodiments, the characteristic is an enablement of across-component linear model (CCLM) coding mode in the current videoblock.

In some embodiments, the characteristic is a size of the current videoblock.

In some embodiments, the characteristic is a splitting depth of apartitioning process applied to the current video block. In an example,the partitioning process is a quadtree (QT) partitioning process, abinary tree (BT) partitioning process or a ternary tree (TT)partitioning process.

In some embodiments, the characteristic is a color format or a colorcomponent of the current video block.

In some embodiments, the characteristic excludes an intra predictionmode of the current video block and an index of a multiple transformselection (MTS) process.

FIG. 22F shows a flowchart of an exemplary method for video processing.The method 2250 includes, at step 2252, making a decision, based on acharacteristic of a current video block, regarding a selectiveapplication of an inverse reduced secondary transform (RST) process onthe current video block.

The method 2250 includes, at step 2254, performing, based on thedecision, a conversion between the current video block and a bitstreamrepresentation of a video comprising the current video block.

In some embodiments, the characteristic is a coded block flag of acoding group of the current video block. In an example, the inverse RSTprocess is not applied, and wherein the coded block flag of a top-leftcoding group is zero. In another example, the inverse RST process is notapplied, and wherein coded block flags for a first and a second codinggroup in a scanning order of the current video block are zero.

In some embodiments, the characteristic is a height (M) or a width (N)of the current video block. In an example, the inverse RST process isnot applied, and wherein (i) M=8 and N=4, or (ii) M=4 and N=8.

As further described in the listing in the previous section (e.g., items27 to 29), in some embodiments, a method of video processing includesdetermining, for a conversion between a coded representation of acurrent video block comprising sub-blocks and the current video block, azero-out region applied for the conversion of the sub-blocks based on acoding condition; and performing the conversion based on thedetermining.

In this method, the coding condition comprises a size of the sub-blocks(see, e.g., item 27).

In this method, the coding condition includes a size of a secondarytransform used during the conversion (see, e.g., item 28).

In some embodiments, the conversion may be performed using reduced sizetransform matrices (see, e.g., item 29).

5 Example Implementations of the Disclosed Technology

In the following exemplary embodiments, the changes on top of JVET-N0193are highlighted in bold and Italic. Deleted texts are marked with doublebrackets (e.g., [[a]] denotes the deletion of the character “a”).

5.1 Embodiment #1

Signaling of RST index is dependent on number of non-zero coefficientswithin a sub-region of the block, instead of the whole block.

7.3.6.11 Residual Coding Syntax

Descriptor residual_coding( x0, y0, log2TbWidth, log2TbHeight, cIdx ) { if( ( tu_mts_idx[ x0 ][ y0 ] > 0 ||    ( cu_sbt_flag && log2TbWidth < 6&& log2TbHeight < 6 ) )    && cIdx = = 0 && log2TbWidth > 4 )  log2TbWidth = 4  else   log2TbWidth = Min( log2TbWidth, 5 )  if(tu_mts_idx[ x0 ][ y0 ] > 0 ||    ( cu_sbt_flag && log2TbWidth < 6 &&log2TbHeight < 6 ) )    && cIdx = = 0 && log2TbHeight > 4 )  log2TbHeight = 4  else   log2TbHeight = Min( log2TbHeight, 5 )  if(log2TbWidth > 0 )   last_sig_coeff_x_prefix ae(v)  if( log2TbHeight > 0)   last_sig_coeff_y_prefix ae(v)  if( last_sig_coeff_x_prefix > 3 )  last_sig_coeff_x_suffix ae(v)  if( last_sig_coeff_y_prefix > 3 )  last_sig_coeff_y_suffix ae(v)  log2SbW = ( Min( log2TbWidth,log2TbHeight ) < 2 ? 1 : 2 )  log2SbH = log2SbW  if ( log2TbWidth < 2 &&cIdx = = 0 ) {   log2SbW = log2TbWidth   log2SbH = 4 − log2SbW  } elseif ( log2TbHeight < 2 && cIdx = = 0 ) {   log2SbH = log2TbHeight  log2SbW = 4 − log2SbH  }  numSbCoeff = 1 << ( log2SbW + log2SbH ) lastScanPos = numSbCoeff  lastSubBlock = ( 1 << ( log2TbWidth +log2TbHeight − ( log2SbW + log2SbH ) ) ) − 1  do {   if( lastScanPos = =0 ) {    lastScanPos = numSbCoeff    lastSubBlock− −   }   lastScanPos−−   xS = DiagScanOrder[ log2TbWidth − log2SbW ][ log2TbHeight − log2SbH]         [ lastSubBlock ][ 0 ]   yS = DiagScanOrder[ log2TbWidth −log2SbW ][ log2TbHeight − log2SbH ]         [ lastSubBlock ][ 1 ]   xC =( xS << log2SbW ) + DiagScanOrder[ log2SbW ][ log2SbH ][ lastScanPos ][0 ]   yC = ( yS << log2SbH ) + DiagScanOrder[ log2SbW ][ log2SbH ][lastScanPos ][ 1 ]  } while( ( xC != LastSignificantCoeffX ) || ( yC !=LastSignificantCoeffY ) )  QState = 0  for( i = lastSubBlock; i >= 0; i−− ) {   startQStateSb = QState   xS = DiagScanOrder[ log2TbWidth −log2SbW ][ log2TbHeight − log2SbH ]         [ lastSubBlock ][ 0 ]   yS =DiagScanOrder[ log2TbWidth − log2SbW ][ log2TbHeight − log2SbH ]        [ lastSubBlock ][ 1 ]   inferSbDcSigCoeffFlag = 0   if( ( i <lastSubBlock ) && ( i > 0 ) ) {    coded_sub_block_flag[ xS ][ yS ]ae(v)    inferSbDcSigCoeffFlag = 1   }   firstSigScanPosSb = numSbCoeff  lastSigScanPosSb = −1   remBinsPass1 = ( ( log2SbW + log2SbH ) < 4 ? 8: 32 )   firstPosMode0 = ( i = = lastSubBlock ? lastScanPos : numSbCoeff− 1 )   firstPosMode1 = −1   for( n = firstPosMode0; n >= 0 &&remBinsPass1 >= 4; n− − ) {    xC = ( xS << log2SbW ) + DiagScanOrder[log2SbW ][ log2SbH ][ n ][ 0 ]    yC = ( yS << log2SbH ) +DiagScanOrder[ log2SbW ][ log2SbH ][ n ][ 1 ]    if(coded_sub_block_flag[ xS ][ yS ] && ( n > 0 || !inferSbDcSigCoeffFlag )&&     ( xC != LastSignificantCoeffX || yC != Last SignificantCoeffY ) ){     sig_coeff_flag[ xC ][ yC ] ae(v)     remBinsPass1− −     if(sig_coeff_flag[ xC ][ yC ] )      inferSbDcSigCoeffFlag = 0    }    if(sig_coeff_flag[ xC ][ yC ] ) {     if( !transform_skip_flag[ x0 ][ y0 ]) {        if (i = 0 ∥ (i == 1 && (log2TbWidth+log2TbHeight==5)))       numSigCoeff++      if( ( ( ( log2TbWidth == 2 && log2TbHeight ==2 ) || ( log2TbWidth == 3 &&       log2TbHeight == 3 ) ) && n >= 8 && i== 0 ) || ( ( log2TbWidth >= 3 &&       log2TbHeight >= 3 && ( i == 1 ||i == 2 ) ) ) ) {       numZeroOutSigCoeff++      }     }    abs_level_gt1_flag[ n ] ae(v)     remBinsPass1− −     if(abs_level_gt1_flag[ n ] ) {      par_level_flag[ n ] ae(v)     remBinsPass1− −      abs_level_gt3_flag[ n ] ae(v)     remBinsPass1− −     }     if( lastSigScanPosSb = = −1 )     lastSigScanPosSb = n     firstSigScanPosSb = n    } ...  } }Alternatively, the condition may be replaced by:

 

 

 

 

5.2 Embodiment #2

RST may not be invoked according to coded block flags of certain CGs.

8.7.4. Transformation Process for Scaled Transform Coefficients

8.7.4.1 General

Inputs to this process are:

-   -   a luma location (xTbY, yTbY) specifying the top-left sample of        the current luma transform block relative to the top-left luma        sample of the current picture,

    -   a variable nTbW specifying the width of the current transform        block,

    -   a variable nTbH specifying the height of the current transform        block,

    -   a variable cIdx specifying the colour component of the current        block,

    -   an (nTbW)×(nTbH) array d[x] [y] of scaled transform coefficients        with x=0 . . . nTbW−1, y=0 . . . nTbH−1.        Output of this process is the (nTbW)×(nTbH) array r[x] [y] of        residual samples with x=0 . . . nTbW−1, y=0 . . . nTbH−1.

    -   

    -   

    -           st_idx[xTbY] [yTbY] is not equal to 0, the following applies:

    -   1. The variables nStSize, log 2StSize, numStX, numStY, and        nonZeroSize are derived as follows:        -   If both nTbW and nTbH are greater than or equal to 8, log            2StSize is set to 3 and nStOutSize is set to 48.        -   Otherwise, log 2StSize is set to 2 and nStOutSize is set to            16.        -   nStSize is set to (1<<log 2StSize).        -   If nTbH is equal to 4 and nTbW is greater than 8, numStX set            equal to 2.        -   Otherwise, numStX set equal to 1.        -   If nTbW is equal to 4 and nTbH is greater than 8, numStY set            equal to 2.        -   Otherwise, numStY set equal to 1.        -   If both nTbW and nTbH are equal to 4 or both nTbW and nTbH            are equal to 8, nonZeroSize is set equal to 8.        -   Otherwise, nonZeroSize set equal to 16.

    -   2. For xSbIdx=0 . . . numStX−1 and ySbIdx=0 . . . numStY−1, the        following applies:        -   The variable array u[x] with x=0 . . . nonZeroSize−1 are            derived as follows:            xC=(xSbIdx<<log 2StSize)+DiagScanOrder[log 2StSize][log            2StSize][x][0]            yC=(ySbIdx<<log 2StSize)+DiagScanOrder[log 2StSize][log            2StSize][x][1]            u[x]=d[xC][yC]

    -   u[x] with x=0 . . . nonZeroSize−1 is transformed to the variable        array v[x] with x=0 . . . nStOutSize−1 by invoking the        one-dimensional transformation process as specified in clause        8.7.4.4 with the transform input length of the scaled transform        coefficients nonZeroSize, the transform output length nStOutSize        the list u[x] with x=0 . . . nonZeroSize−1, the index for        transform set selection stPredModeIntra, and the index for        transform selection in a transform set st_idx[xTbY] [yTbY] as        inputs, and the output is the list v[x] with x=0 . . .        nStOutSize−1. The variable stPredModeIntra is set to the        predModeIntra specified in clause 8.4.4.2.1.        -   The array d[(xSbIdx<<log 2StSize)+x] [(ySbIdx<<log            2StSize)+y] with x=0 . . . nStSize 1, y=0 . . . nStSize−1            are derived as follows:            -   If stPredModeIntra is less than or equal to 34, or equal                to INTRA_LT_CCLM, INTRA_T_CCLM, or INTRA_L_CCLM, the                following applies:                d[(xSbIdx<<log 2StSize)+x][(ySbIdx<<log                2StSize)+y]=(y<4)?v[x+(y<<log                2StSize)]:((x<4)?v[32+x+((y−4)<<2)]: d[(xSbIdx<<log                2StSize)+x][(ySbIdx<<log 2StSize)+y])            -   Otherwise, the following applies:                d[(xSbIdx<<log 2StSize)+x][(ySbIdx<<log                2StSize)+y]=(y<4)?v[y+(x<<log                2StSize)]:((x<4)?v[32+(y−4)+(x<<2)]: d[(xSbIdx<<log                2StSize)+x][(ySbIdx<<log 2StSize)+y])                The variable implicitMtsEnabled is derived as follows:

    -   If sps_mts_enabled_flag is equal to 1 and one of the following        conditions is true, implicitMtsEnabled is set equal to 1:        -   IntraSubPartitionsSplitType is not equal to ISP_NO_SPLIT        -   cu_sbt_flag is equal to 1 and Max(nTbW, nTbH) is less than            or equal to 32        -   sps_explicit_mts_intra_enabled_flag and            sps_explicit_mts_inter_enabled_flag are both equal to 0 and            CuPredMode[xTbY] [yTbY] is equal to MODE_INTRA

    -   Otherwise, implicitMtsEnabled is set equal to 0.        The variable trTypeHor specifying the horizontal transform        kernel and the variable trTypeVer specifying the vertical        transform kernel are derived as follows:

    -   If cIdx is greater than 0, trTypeHor and trTypeVer are set equal        to 0.

    -   Otherwise, if implicitMtsEnabled is equal to 1, the following        applies:        -   If IntraSubPartitionsSplitType is not equal to ISP_NO_SPLIT,            trTypeHor and trTypeVer are specified in Table 8-15            depending on intraPredMode.        -   Otherwise, if cu_sbt_flag is equal to 1, trTypeHor and            trTypeVer are specified in Table 8-14 depending on            cu_sbt_horizontal_flag and cu_sbt_pos_flag.        -   Otherwise (sps_explicit_mts_intra_enabled_flag and            sps_explicit_mts_inter_enabled_flag are equal to 0),            trTypeHor and trTypeVer are derived as follows:            trTypeHor=(nTbW>=4&& nTbW<=16&& nTbW<=nTbH)?1:0  (8-1029)            trTypeVer=(nTbH>=4&& nTbH<=16&& nTbH<=nTbW)?1:0  (8-1030)

    -   Otherwise, trTypeHor and trTypeVer are specified in Table 8-13        depending on tu_mts_idx[xTbY] [yTbY].        The variables nonZeroW and nonZeroH are derived as follows:        nonZeroW=Min(nTbW,(trTypeHor>0)?16:32)  (8-1031)        nonZeroH=Min(nTbH,(trTypeVer>0)?16:32)  (8-1032)        The (nTbW)×(nTbH) array r of residual samples is derived as        follows:

-   1. When nTbH is greater than 1, each (vertical) column of scaled    transform coefficients d[x] [y] with x=0 . . . nonZeroW−1, y=0 . . .    nonZeroH−1 is transformed to e[x] [y] with x=0 . . . nonZeroW−1, y=0    . . . nTbH−1 by invoking the one-dimensional transformation process    as specified in clause 8.7.4.2 for each column x=0 . . . nonZeroW−1    with the height of the transform block nTbH, the non-zero height of    the scaled transform coefficients nonZeroH, the list d[x] [y] with    y=0 . . . nonZeroH−1 and the transform type variable trType set    equal to trTypeVer as inputs, and the output is the list e[x] [y]    with y=0 . . . nTbH−1.

-   2. When nTbH and nTbW are both greater than 1, the intermediate    sample values g[x] [y] with x=0 . . . nonZeroW−1, y=0 . . . nTbH−1    are derived as follows:    g[x][y]=Clip3(CoeffMin,CoeffMax,(e[x][y]+64)>>7)  (8-1033)    When nTbW is greater than 1, each (horizontal) row of the resulting    array g[x] [y] with x=0 . . . nonZeroW−1, y=0 . . . nTbH−1 is    transformed to r[x] [y] with x=0 . . . nTbW−1, y=0 . . . nTbH−1 by    invoking the one-dimensional transformation process as specified in    clause 8.7.4.2 for each row y=0 . . . nTbH−1 with the width of the    transform block nTbW, the non-zero width of the resulting array g[x]    [y] nonZeroW, the list g[x] [y] with x=0 . . . nonZeroW−1 and the    transform type variable trType set equal to trTypeHor as inputs, and    the output is the list r[x] [y] with x=0 . . . nTbW−1.

5.3 Embodiment #3

Context modeling of RST index is revised.

5.3.1 Alternative #1

9.5.4.2.8 Derivation Process of ctxInc for the Syntax Element st_idx

Inputs to this process are the colour component index cIdx, the luma orchroma location (x0, y0) specifying the top-left sample of the currentluma or chroma coding block relative to the top-left sample of thecurrent picture depending on cIdx, the tree type treeType, the lumaintra prediction mode IntraPredModeY[x0] [y0] as specified in clause8.4.2, the syntax element intra_chroma_pred_mode[x0] [y0] specifying theintra prediction mode for chroma samples as specified in clause 7.4.7.5,

and the multiple transform selection index tu_mts_idx[x0] [y0].Output of this process is the variable ctxInc.The variable intraModeCtx is derived as follows:If cIdx is equal to 0, intraModeCtx is derived as follows:intraModeCtx=(IntraPredModeY[x0][y0]<=1)?1:0Otherwise (cIdx is greater than 0), intraModeCtx is derived as follows:intraModeCtx=(intra_chroma_pred_mode[x0][y0]>=4)?1:0The variable mtsCtx is derived as follows:mtsCtx=(

tu_mts_idx[x0][y0]==0:

&& treeType!=SINGLE_TREE)?1:0The variable ctxInc is derived as follows:ctxInc=(binIdx<<1)+intraModeCtx+(mtsCtx<<2)5.3.2 Alternative #2

Binarization Syntax element Process Input parameters Syntax . . . . . .. . . structure st_idx[ ][ ] TR cMax = 2, cRiceParam = 0

TABLE 9-15 Assignment of ctxInc to syntax elements with context codedbins Syntax binIdx element 0 1 2 3 4 >=5 . . . . . . . . . . . . . . . .. . . . . st_idx[ ][ ] 0[[, 1, 4, 5]] 2[[, 3, 6, 7]] na na na na (clause9.5.4.2.8) (clause 9.5.4.2.8) . . . . . . . . . . . . . . . . . . . . .[[9.5.4.2.8 Derivation Process of ctxInc for the Syntax Element St_IdxInputs to this process are the colour component index cIdx, the luma orchroma location (x0, y0) specifying the top-left sample of the currentluma or chroma coding block relative to the top-left sample of thecurrent picture depending on cIdx, the tree type treeType, the lumaintra prediction mode IntraPredModeY[x0] [y0] as specified in clause8.4.2, the syntax element intra_chroma_pred_mode[x0] [y0] specifying theintra prediction mode for chroma samples as specified in clause 7.4.7.5,and the multiple transform selection index tu_mts_idx[x0] [y0].Output of this process is the variable ctxInc.The variable intraModeCtx is derived as follows:If cIdx is equal to 0, intraModeCtx is derived as follows:intraModeCtx=(IntraPredModeY[x0][y0]<=1)?1:0Otherwise (cIdx is greater than 0), intraModeCtx is derived as follows:intraModeCtx=(intra_chroma_pred_mode[x0][y0]>=4)?1:0The variable mtsCtx is derived as follows:mtsCtx=(tu_mts_idx[x0][y0]==0&& treeType!=SINGLE_TREE)?1:0The variable ctxInc is derived as follows:ctxInc=(binIdx<<1)+intraModeCtx+(mtsCtx<<2)]]

Corresponding to bullets 7. c and 7.d.

7.3.7.11 Residual Coding Syntax

Descriptor residual_coding( x0, y0, log2TbWidth, log2TbHeight, cIdx ) {...  if( coded_sub_block_flag[ xS ][ yS ] && ( n > 0 ||!inferSbDcSigCoeffFlag ) &&   ( xC != LastSignificantCoeffX || yC !=Last SignificantCoeffY ) ) {   sig_coeff_flag[ xC ][ yC ] ae(v)  remBinsPass1−−   if( sig_coeff_flag[ xC ][ yC ] )   inferSbDcSigCoeffFlag = 0  }  if( sig_coeff_flag[ xC ][ yC ] ) {  if( !transform_skip_flag[ x0 ][ y0 ] ) {   if(xC<4 && yC<4)  numSigCoeff++    if( ( ( ( log2TbWidth == 2 && log2TbHeight == 2 ) ||( log2TbWidth == 3 && log2TbHeight == 3 ) ) && n >= 8 && i == 0 ) || ( (log2TbWidth >= 3 && log2TbHeight >= 3 && ( i == 1 || i == 2 ) ) ) ) {    numZeroOutSigCoeff++    }   }   abs_level_gt1_flag[ n ] ae(v) ...In an alternative example, the following may apply:

Descriptor residual_coding( x0, y0, log2TbWidth, log2TbHeight, cIdx ) {...  if( coded_sub_block_flag[ xS ][ yS ] && ( n > 0 | |!inferSbDcSigCoeffFlag ) &&   ( xC != LastSignificantCoeffX | | yC !=Last SignificantCoeffY ) ) {   sig_coeff_flag[ xC ][ yC ] ae(v)  remBinsPass1−−   if( sig_coeff_flag[ xC ][ yC ] )   inferSbDcSigCoeffFlag = 0  }  if( sig_coeff_flag[ xC ][ yC ] ) {  if( !transform_skip_flag[ x0 ][ y0 ] ) {   if(xC<SigRangeX &&yC<SigRangeY)    numSigCoeff++     if( ( ( ( log2TbWidth == 2 &&log2TbHeight == 2 ) | | ( log2TbWidth == 3 && log2TbHeight == 3 ) ) &&n >= 8 && i == 0 ) | | ( ( log2TbWidth >= 3 && log2TbHeight >= 3 && ( i== 1 | | i == 2 ) ) ) ) {      numZeroOutSigCoeff++ e   }   abs_level_gt1_flag[ n ] ae(v) ...

Corresponding to Bullet 19.

7.3.6.5 Coding unit syntax

Descriptor coding_unit( x0, y0, cbWidth, cbHeight, treeType ) { ...  if(!pcm_flag[ x0 ][ y0 ] ) {   if( CuPredMode[ x0 ][ y0 ] != MODE_INTRA &&merge_flag[ x0 ][ y0 ] = = 0 )    cu_cbf ae(v)   if( cu_cbf ) {    if(CuPredMode[ x0 ][ y0 ] = = MODE_INTER && sps_sbt_enabled_flag &&    !ciip_flag[ x0 ][ y0 ] ) {     if( cbWidth <= MaxSbtSize && cbHeight<= MaxSbtSize ) {      allowSbtVerH = cbWidth >= 8      allowSbtVerQ =cbWidth >= 16      allowSbtHorH = cbHeight >= 8      allowSbtHorQ =cbHeight >= 16      if( allowSbtVerH || allowSbtHorH || allowSbtVerQ ||allowSbtHorQ )       cu_sbt_flag ae(v)     }     if( cu_sbt_flag ) {     if( ( allowSbtVerH || allowSbtHorH ) && ( allowSbtVerQ ||allowSbtHorQ) )       cu_sbt_quad_flag ae(v)      if( ( cu_sbt_quad_flag&& allowSbtVerQ && allowSbtHorQ ) ||        ( !cu_sbt_quad_flag &&allowSbtVerH && allowSbtHorH ) )       cu_sbt_horizontal_flag ae(v)     cu_sbt_pos_flag ae(v)     }    }    numZeroOutSigCoeff = 0   transform_tree( x0, y0, cbWidth, cbHeight, treeType )    if( Min(cbWidth, cbHeight ) >= 4 && sps_st_enabled_flag == 1 && CuPredMode[ x0][ y0 ] = = MODE_INTRA && IntraSubPartitionsSplitType == ISP_NO_SPLIT ){     if( ( numSigCoeff > [[( ( treeType == SINGLE_TREE ) ? 2 : ]]1 [[)]] ) && numZeroOutSigCoeff == 0 ) {      st_idx[ x0 ][ y0 ] ae(v)     }   }   }  } }

Corresponding to bullet 20.

7.3.6.5 Coding Unit Syntax

Descriptor coding_unit( x0, y0, cbWidth, cbHeight, treeType ) { ...  if(!pcm_flag[ x0 ][ y0 ] ) {   if( CuPredMode[ x0 ][ y0 ] != MODE_INTRA &&merge_flag[ x0 ][ y0 ] = = 0 )    cu_cbf ae(v)   if( cu_cbf ) {    if(CuPredMode[ x0 ][ y0 ] = = MODE_INTER && sps_sbt_enabled_flag &&    !ciip_flag[ x0 ][ y0 ] ) {     if( cbWidth <= MaxSbtSize && cbHeight<= MaxSbtSize ) {      allowSbtVerH = cbWidth >= 8      allowSbtVerQ =cbWidth >= 16      allowSbtHorH = cbHeight >= 8      allowSbtHorQ =cbHeight >= 16      if( allowSbtVerH | | allowSbtHorH | | allowSbtVerQ || allowSbtHorQ )       cu_sbt_flag ae(v)     }     if( cu_sbt_flag ) {     if( ( allowSbtVerH | | allowSbtHorH ) && ( allowSbtVerQ | |allowSbtHorQ) )       cu_sbt_quad_flag ae(v)      if( ( cu_sbt_quad_flag&& allowSbtVerQ && allowSbtHorQ ) | |        ( !cu_sbt_quad_flag &&allowSbtVerH && allowSbtHorH ) )       cu_sbt_horizontal_flag ae(v)     cu_sbt_pos_flag ae(v)     }    }    numZeroOutSigCoeff = 0   transform_tree( x0, y0, cbWidth, cbHeight, treeType )    if( Min(cbWidth, cbHeight ) >= 4 && sps_st_enabled_flag == 1 && CuPredMode[ x0][ y0 ] = = MODE_INTRA && IntraSubPartitionsSplitType == ISP_NO_SPLIT ){     if( ( numSigCoeff > ( ( treeType == SINGLE_TREE ) ? 2 : 1 ) ) &&numZeroOutSigCoeff == 0 && cbWidth<=MaxTbSizeY && cbHeight<=MaxTbSizeY ){       st_idx[ x0 ][ y0 ] ae(v)     }    }   }  } }

Corresponding to bullet 21.

7.3.7.11 Residual Coding Syntax

Descriptor residual_coding( x0, y0, log2TbWidth, log2TbHeight, cIdx ) {...  if( coded_sub_block_flag[ xS ][ yS ] && ( n > 0 ||!inferSbDcSigCoeffFlag ) &&   ( xC != LastSignificantCoeffX || yC !=Last SignificantCoeffY ) ) {   sig_coeff_flag[ xC ][ yC ] ae(v)  remBinsPass1− −   if( sig_coeff_flag[ xC ][ yC ] )   inferSbDcSigCoeffFlag = 0  }  if( sig_coeff_flag[ xC ][ yC ]&& x0 = =CbX[x0][y0]&&y0 = = CbU[x0][y0]){   if( !transform_skip_flag[ x0 ][ y0 ]) {   numSigCoeff++    if( ( ( ( log2TbWidth == 2 && log2TbHeight == 2 )|| ( log2TbWidth == 3 && log2TbHeight == 3 ) ) && n >= 8 && i == 0 ) ||( ( log2TbWidth >= 3 && log2TbHeight>= 3 && ( i == 1 || i == 2 ) ) ) ) {    numZeroOutSigCoeff++    }   }   abs_level_gt1_flag[ n ] ae(v) ...

The RST transform set index is derived from default modes assigned tonon-normal intra prediction modes. The newly added parts are highlightedin bold and Italic and the deleted parts are highlighted in are markedwith double brackets (e.g., [[a]] denotes the deletion of the character“a”).

8.7.4.1 General

Inputs to this process are:

-   -   a luma location (xTbY, yTbY) specifying the top-left sample of        the current luma transform block relative to the top-left luma        sample of the current picture,    -   a variable nTbW specifying the width of the current transform        block,    -   a variable nTbH specifying the height of the current transform        block,    -   a variable cIdx specifying the colour component of the current        block,    -   an (nTbW)×(nTbH) array d[x] [y] of scaled transform coefficients        with x=0 . . . nTbW−1, y=0 . . . nTbH−1.        Output of this process is the (nTbW)×(nTbH) array r[x] [y] of        residual samples with x=0 . . . nTbW−1, y=0 . . . nTbH−1.        When lfnst_idx[xTbY] [yTbY] is not equal to 0 and both nTbW and        nTbH are greater than or equal to 4, the following applies:    -   The variables predModeIntra, nLfnstOutSize, log 2LfnstSize,        nLfnstSize, and nonZeroSize are derived as follows:        predModeIntra=(cIdx==0)?IntraPredModeY[xTbY][yTbY]:IntraPredModeC[xTbY][yTbY]  (8-965)        nLfnstOutSize=(nTbW>=8&& nTbH>=8)?48:16  (8-966)        log 2LfnstSize=(nTbW>=8 && nTbH>=8)?3:2  (8-967)        nLfnstSize=1<<log 2LfnstSize  (8-968)        nonZeroSize=((nTbW==4&& nTbH==4)(nTbW==8&&        nTbH==8))?8:16  (8-969)    -   When intra_mip_flag[xTbComp] [yTbComp] is equal to 1 and cIdx is        equal to 0, predModeIntra is set equal to INTRA_PLANAR.    -   When predModeIntra is equal to either INTRA_LT_CCLM,        INTRA_L_CCLM, or INTRA_T_CCLM, predModeIntra        [[is set equal to IntraPredModeY[xTbY+nTbW/2] [yTbY+nTbH/2].]]        -   

        -   

        -       -   The wide angle intra prediction mode mapping process as        specified in clause 8.4.5.2.6 is invoked with predModeIntra,        nTbW, nTbH and cIdx as inputs, and the modified predModeIntra as        output.        Alternatively, the followings may apply:    -   When predModeIntra is equal to either INTRA_LT_CCLM,        INTRA_L_CCLM, or INTRA_T_CCLM, predModeIntra        [[is set equal to IntraPredModeY[xTbY+nTbW/2] [yTbY+nTbH/2].]]        -   

        -   

        -   

The RST transform set index is derived from default modes assigned tonon-normal intra prediction modes, and dependent on color format. Thenewly added parts are highlighted in bold and Italic and the deletedparts are highlighted in are marked with double brackets (e.g., [[a]]denotes the deletion of the character “a”).

8.7.42 General

Inputs to this process are:

-   -   a luma location (xTbY, yTbY) specifying the top-left sample of        the current luma transform block relative to the top-left luma        sample of the current picture,    -   a variable nTbW specifying the width of the current transform        block,    -   a variable nTbH specifying the height of the current transform        block,    -   a variable cIdx specifying the colour component of the current        block,    -   an (nTbW)×(nTbH) array d[x] [y] of scaled transform coefficients        with x=0 . . . nTbW−1, y=0 . . . nTbH−1.        Output of this process is the (nTbW)×(nTbH) array r[x] [y] of        residual samples with x=0 . . . nTbW−1, y=0 . . . nTbH−1.        When lfnst_idx[xTbY] [yTbY] is not equal to 0 and both nTbW and        nTbH are greater than or equal to 4, the following applies:    -   The variables predModeIntra, nLfnstOutSize, log 2LfnstSize,        nLfnstSize, and nonZeroSize are derived as follows:        predModeIntra=(cIdx==0)?IntraPredModeY[xTbY][yTbY]:IntraPredModeC[xTbY][yTbY]  (8-965)        nLfnstOutSize=(nTbW>=8&& nTbH>=8)?48:16  (8-966)        log 2LfnstSize=(nTbW>=8&& nTbH>=8)?3:2  (8-967)        nLfnstSize=1<<log 2LfnstSize  (8-968)        nonZeroSize=((nTbW==4&& nTbH==4)(nTbW==8&&        nTbH==8))?8:16  (8-969)    -   When intra_mip_flag[xTbComp] [yTbComp] is equal to 1 and cIdx is        equal to 0, predModeIntra is set equal to INTRA_PLANAR.    -   When predModeIntra is equal to either INTRA_LT_CCLM,        INTRA_L_CCLM, or INTRA_T_CCLM, predModeIntra        [[is set equal to IntraPredModeY[xTbY+nTbW/2] [yTbY+nTbH/2].]]        -   

        -   

        -   

        -       -   The wide angle intra prediction mode mapping process as        specified in clause 8.4.5.2.6 is invoked with predModeIntra,        nTbW, nTbH and cIdx as inputs, and the modified predModeIntra as        output.

TABLE 8-3 Specification of the 4:2:2 mapping process from chroma intraprediction mode X to mode Y when chroma_format_idc is equal to 2 mode X0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 mode Y 0 1 61 62 63 64 65 662 3 4 6 8 10 12 13 14 16 mode X 18 19 20 21 22 23 24 25 26 27 28 29 3031 32 33 34 35 mode Y 18 20 22 23 24 26 28 30 32 33 34 35 36 37 38 39 4041 mode X 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 mode Y42 43 44 44 44 45 46 46 46 47 48 48 48 49 50 51 52 52 mode X 54 55 56 5758 59 60 61 62 63 64 65 66 mode Y 52 53 54 54 54 55 56 56 56 57 58 59 60Alternatively, the followings may apply:

-   -   When predModeIntra is equal to either INTRA_LT_CCLM,        INTRA_L_CCLM, or INTRA_T_CCLM, predModeIntra        [[[is set equal to IntraPredModeY[xTbY+nTbW/2] [yTbY+nTbH/2].]]        -   

        -   

        -   

        -   

In this embodiment, the center luma sample of corresponding luma regionof current chroma block is checked whether it is coded with MIP or IBCor palette mode; and the setting of DM is also based on the center lumasample.

The newly added parts are highlighted in bold and Italic and the deletedparts are highlighted in are marked with double brackets (e.g., [[a]]denotes the deletion of the character “a”).

8.4.3 Derivation Process for Chroma Intra Prediction Mode

Input to this process are:

-   -   a luma location (xCb, yCb) specifying the top-left sample of the        current chroma coding block relative to the top-left luma sample        of the current picture,    -   a variable cbWidth specifying the width of the current coding        block in luma samples,    -   a variable cbHeight specifying the height of the current coding        block in luma samples.        In this process, the chroma intra prediction mode        IntraPredModeC[xCb] [yCb] is derived.        The corresponding luma intra prediction mode lumaIntraPredMode        is derived as follows:    -   If intra_mip_flag[xCb+        ] [yCb+        ] is equal to 1, lumaIntraPredMode is set equal to INTRA_PLANAR.    -   Otherwise, if CuPredMode[0] [xCb+        ] [yCb+        ] is equal to MODE_IBC or MODE_PLT, lumaIntraPredMode is set        equal to INTRA_DC.    -   Otherwise, lumaIntraPredMode is set equal to        IntraPredModeY[xCb+cbWidth/2] [yCb+cbHeight/2].        The chroma intra prediction mode IntraPredModeC[xCb] [yCb] is        derived using cclm_mode_mode_flag, cclm_mode_idx,        intra_chroma_pred_mode and lumaIntraPredMode as specified in        Table 8-2.

TABLE 8-2 Specification of IntraPredModeC[ xCb ][ yCb ] depending oncclm_mode_flag, cclm_mode_idx, intra_chroma_pred_mode andlumaIntraPredMode lumaIntraPredMode X (0 <= cclm_mode_flag cclm_mode_idxintra_chroma_pred_mode 0 50 18 1 X <= 66) 0 — 0 66 0 0 0 0 0 — 1 50 6650 50 50 0 — 2 18 18 66 18 18 0 — 3 1 1 1 66 1 0 — 4 0 50 18 1 X 1 0 —81 81 81 81 81 1 1 — 82 82 82 82 82 1 2 — 83 83 83 83 83When chroma_format_idc is equal to 2, the chroma intra prediction mode Yis derived using the chroma intra prediction mode X in Table 8-2 asspecified in Table 8-3, and the chroma intra prediction mode X is is setequal to the chroma intra prediction mode Y afterwards.

In this embodiment, the top-left luma sample of corresponding lumaregion of current chroma block is checked whether it also based on thecenter luma sample.

The newly added parts are highlighted in bold and Italic and the deletedparts are highlighted in are marked with double brackets (e.g., [[a]]denotes the deletion of the character “a”).

8.4.3 Derivation Process for Chroma Intra Prediction Mode

Input to this process are:

-   -   a luma location (xCb, yCb) specifying the top-left sample of the        current chroma coding block relative to the top-left luma sample        of the current picture,    -   a variable cbWidth specifying the width of the current coding        block in luma samples,    -   a variable cbHeight specifying the height of the current coding        block in luma samples.        In this process, the chroma intra prediction mode        IntraPredModeC[xCb] [yCb] is derived.        The corresponding luma intra prediction mode lumaIntraPredMode        is derived as follows:    -   If intra_mip_flag[xCb] [yCb] is equal to 1, lumaIntraPredMode is        set equal to INTRA_PLANAR.    -   Otherwise, if CuPredMode[0] [xCb] [yCb] is equal to MODE_IBC or        MODE_PLT, lumaIntraPredMode is set equal to INTRA_DC.    -   Otherwise, lumaIntraPredMode is set equal to        IntraPredModeY[xCb[[+cbWidth/2]] ] [yCb[[+cbHeight/2]] ].        The chroma intra prediction mode IntraPredModeC[xCb] [yCb] is        derived using cclm_mode_mode_flag, cclm_mode_idx,        intra_chroma_pred_mode and lumaIntraPredMode as specified in        Table 8-2.

TABLE 8-2 Specification of IntraPredModeC[ xCb ][ yCb ] depending oncclm_mode_flag, cclm_mode_idx, intra_chroma_pred_mode andlumaIntraPredMode lumaIntraPredMode X (0 <= cclm_mode_flag cclm_mode_idxintra_chroma_pred_mode 0 50 18 1 X <= 66) 0 — 0 66 0 0 0 0 0 — 1 50 6650 50 50 0 — 2 18 18 66 18 18 0 — 3 1 1 1 66 1 0 — 4 0 50 18 1 X 1 0 —81 81 81 81 81 1 1 — 82 82 82 82 82 1 2 — 83 83 83 83 83When chroma_format_idc is equal to 2, the chroma intra prediction mode Yis derived using the chroma intra prediction mode X in Table 8-2 asspecified in Table 8-3, and the chroma intra prediction mode X is is setequal to the chroma intra prediction mode Y afterwards.

This embodiment shows an example on the context modeling of RST (a.k.aLFNST) index. The deleted parts are highlighted in are marked withdouble brackets (e.g., [[a]] denotes the deletion of the character “a”).

TABLE 9-82 Assignment of ctxInc to syntax elements with context codedbins lfnst_idx[ ][ ] [[( tu_mts_idx[ x0 ][ y0 ] == by- na na na na 0 &&)] treeType != pass SINGLE_TREE [[)]] ? 1 : 0Alternatively, the following may apply:

lfnst_idx[ ][ ] 0 bypass na na na na

The zero-out range is revised for RST4×4. The newly added parts on topof JVET-02001 are highlighted in bold and Italic and the deleted partsare highlighted in are marked with double brackets (e.g., [[a]] denotesthe deletion of the character “a”).

7.3.8.11 Residual Coding Syntax

Descriptor residual_coding( x0, y0, log2TbWidth, log2TbHeight, cIdx ) {...  if( lastSubBlock = = 0 && log2TbWidth >= 2 && log2TbHeight >= 2 &&  !transform_skip_flag[ x0 ][ y0 ] && lastScanPos > 0 )   LfnstDcOnly =0  if( ( lastSubBlock > 0 && log2TbWidth >= 2 && log2TbHeight >= 2 ) ||  ( lastScanPos > 7 && (( log2TbWidth = = 2 ∥ log2TbHeight = = 2 ) || (log2TbWidth = = 3 [[ ) ]] &&   log2TbWidth = = log2TbHeight ) ) ))  LfnstZeroOutSigCoeffFlag = 0  QState = 0 ...Alternatively, the following may apply:

Descriptor residual_coding( x0, y0, log2TbWidth, log2TbHeight, cIdx ) {...  if( lastSubBlock = = 0 && log2TbWidth >= 2 && log2TbHeight >= 2 &&  !transform_skip_flag[ x0 ][ y0 ] && lastScanPos > 0 )   LfnstDcOnly =0  if( [[( ]] lastSubBlock > 0 [[&& log2TbWidth >= 2 && log2TbHeight >=2 ) ]] ||   ( lastScanPos > 7 && !(log2TbWidth > 3 && log2TbHeight > 3)[[ ( log2TbWidth = = 2 || log2TbWidth = = 3 ) &&   log2TbWidth = =log2TbHeight ]]) )   LfnstZeroOutSigCoeffFlag = 0  QState = 0 ...8.7.4.1 GeneralInputs to this process are:

-   -   a luma location (xTbY, yTbY) specifying the top-left sample of        the current luma transform block relative to the top-left luma        sample of the current picture,    -   a variable nTbW specifying the width of the current transform        block,    -   a variable nTbH specifying the height of the current transform        block,    -   a variable cIdx specifying the colour component of the current        block,    -   an (nTbW)×(nTbH) array d[x] [y] of scaled transform coefficients        with x=0 . . . nTbW−1, y=0 . . . nTbH−1.        Output of this process is the (nTbW)×(nTbH) array r[x] [y] of        residual samples with x=0 . . . nTbW−1, y=0 . . . nTbH−1.        When lfnst_idx[xTbY] [yTbY] is not equal to 0 and both nTbW and        nTbH are greater than or equal to 4, the following applies:    -   The variables predModeIntra, nLfnstOutSize, log 2LfnstSize,        nLfnstSize, and nonZeroSize are derived as follows:        predModeIntra=(cIdx==0)?IntraPredModeY[xTbY][yTbY]:IntraPredModeC[xTbY][yTbY]  (8-965)        nLfnstOutSize=(nTbW>=8&& nTbH>=8)?48:16  (8-966)        log 2LfnstSize=(nTbW>=8&& nTbH>=8)?3:2  (8-967)        nLfnstSize=1<<log 2LfnstSize  (8-968)        nonZeroSize=((nTbW==4[        |nTbH==4)∥(nTbW==8&& nTbH==8))?8:16  (8-969)    -   When intra_mip_flag[xTbComp] [yTbComp] is equal to 1 and cIdx is        equal to 0, predModeIntra is set equal to INTRA_PLANAR.

The dimension of the transform matrices of RST4×4 is reduced by half.The newly added parts on top of JVET-O2001 are highlighted in bold andItalic and the deleted parts are highlighted in are marked with doublebrackets (e.g., [[a]] denotes the deletion of the character “a”).8.7.4.3 Low Frequency Non-Separable Transformation Matrix DerivationProcessInputs to this process are:

-   -   a variable nTrS specifying the transform output length,    -   a variable predModeIntra specifying the intra prediction mode        for LFNST set selection,    -   a variable lfnstIdx specifying the LFNST index for transform        selection in the selected LFNST set.        Output of this process is the transformation matrix        lowFreqTransMatrix.        The variable lfnstTrSetIdx is specified in Table 8-16 depending        on predModeIntra.

TABLE 8-16 Specification of lfnstTrSetIdx predModeIntra lfnstTrSetIdxpredModeIntra < 0 1 0 <= predModeIntra <= 1 0  2 <= predModeIntra <= 121 13 <= predModeIntra <= 23 2 24 <= predModeIntra <= 44 3 45 <=predModeIntra <= 55 2 56 <= predModeIntra <= 80 1The transformation matrix lowFreqTransMatrix is derived based on nTrS,lfnstTrSetIdx, and lfnstIdx as follows:

-   -   If nTrS is equal to 16, lfnstTrSetIdx is equal to 0, and        lfnstIdx is equal to 1, the following applies:

lowFreqTransMatrix[m][n]= {{108 −44 −15 1 −44 19 7 −1 −11 6 2 −1 0 −1 −10} {−40 −97 56 12 −11 29 −12 −3 18 18 −15 −3 −1 −3 2 1} {25 −31 −1 7 100−16 −29 1 −54 21 14 −4 −7 2 4 0} {−32 −39 −92 51 −6 −16 36 −8 3 22 18−15 4 1 −5 2} {8 −9 33 −8 −16 −102 36 23 −4 38 −27 −55 16 −8 −6} {−25 516 −3 −38 14 11 −3 −97 7 26 1 55 −10 −19 3} {8 9 16 1 37 36 94 −38 −7 3−47 11 −6 −13 −17 10} {2 34 −5 1 −7 24 −25 −3 8 99 −28 −29 6 −43 21 11}[[{−16 −27 −39 −109 6 10 16 24 3 19 10 24 −4 −7 −2 −3} {−9 −10 −34 4 −9−5 −29 5 −33 −26 −96 33 14 4 39 −14} {−13 1 4 −9 −30 −17 −3 −64 −35 1117 19 −86 6 36 14} {8 −7 −5 −15 7 −30 −28 −87 31 4 4 33 61 −5 −17 22}{−2 13 −6 −4 −2 28 −13 −14 −3 37 −15 −3 −2 107 −36 −24} {4 9 11 31 4 916 19 12 33 32 94 12 0 34 −45} {2 −2 8 −16 8 5 28 −17 6 −7 18 −45 40 3697 −8} {0 −2 0 −10 −1 −7 −3 −35 −1 −7 −2 −32 −6 −33 −16 −112}]]},

-   -   Otherwise, if nTrS is equal to 16, lfnstTrSetIdx is equal to 0,        and lfnstIdx is equal to 2, the following applies:

lowFreqTransMatrix[m][n]= {{119 −30 −22 −3 −23 −2 3 2 −16 3 6 0 −3 2 10} {−27 −101 31 17 −47 2 22 3 19 30 −7 −9 5 3 −5 −1} {0 58 22 −15 −102 238 2 10 −13 −5 4 14 −1 −9 0} {23 4 66 −11 22 89 −2 −26 13 −8 −38 −1 −9−20 −2 8} {−19 −5 −89 2 −26 76 −11 −17 20 13 18 −4 1 −15 3 5} {−10 −1 −16 23 25 87 −7 −74 4 39 −5 0 −1 −20 −1} {−17 −28 12 −8 −32 14 −53 −6 −68−67 17 29 2 6 25 4} {1 −24 −23 1 17 −7 52 9 50 −92 −15 27 −15 −10 −6 3}[[{−6 −17 −2 −111 7 −17 8 −42 9 18 16 25 −4 2 −1 11} {9 5 35 0 6 21 −934 44 −3 102 11 −7 13 11 −20} {4 −5 −5 −10 15 19 −2 6 6 −12 −13 6 95 69−29 −24} {−6 −4 −9 −39 1 22 0 102 −19 19 −32 30 −16 −14 −8 −23} {4 −4 78 4 −13 −18 5 0 0 21 22 58 −88 −54 28} {−4 −7 0 −24 −7 0 −25 3 −3 −30 8−76 −34 4 −80 −26} {0 6 0 30 −6 1 −13 −23 1 20 −2 80 −44 37 −68 1} {0 0−1 5 −1 −7 1 −34 −2 3 −6 19 5 −38 11 −115}]]},

-   -   Otherwise, if nTrS is equal to 16, lfnstTrSetIdx is equal to 1,        and lfnstIdx is equal to 1, the following applies:

lowFreqTransMatrix[m][n]= {{−111 39 4 3 44 11 −12 −1 7 −16 −5 2 3 −1 42} {−47 −27 15 −1 −92 43 20 −2 20 39 −16 −5 10 −5 −13 2} {−35 −23 4 4−17 −72 32 6 −59 18 50 −6 0 40 0 −13} {13 93 −27 −4 −48 13 −34 4 −52 111 10 3 16 −3 1} {−11 −27 1 2 −47 −4 −36 10 −2 −85 14 29 −20 −2 57 4} {0−35 32 −2 26 60 −3 −17 −82 1 −30 0 −37 21 3 12} {−17 −46 −92 14 7 −10−39 29 −17 27 −28 17 1 −15 −13 17} {4 −10 −23 4 16 58 −17 26 30 21 67 2−13 59 13 −40} [[{5 −20 32 −5 8 −3 −46 −7 −4 2 −15 24 100 44 0 5} {−4 −138 −18 −7 −42 −63 −6 33 34 −23 15 −65 33 −20 2} {−2 −10 35 −19 5 8 −4414 −25 25 58 17 7 −84 −16 −18} {5 13 18 34 11 −4 18 18 5 58 −3 42 −2 −1085 38} {−5 −7 −34 −83 2 −1 −4 −73 4 20 15 −12 4 −3 44 12} {0 4 −2 −60 59 42 34 5 −14 9 80 −5 13 −38 37} {−1 2 7 −57 3 −7 9 68 −9 6 −49 −20 6 −436 −64} {−1 0 −12 23 1 −4 17 −53 −3 4 −21 72 −4 −8 −3 −83}]]},

-   -   Otherwise, if nTrS is equal to 16, lfnstTrSetIdx is equal to 1,        and lfnstIdx is equal to 2, the following applies:

lowFreqTransMatrix[m][n]= {{88 −55 6 −3 −66 27 9 −2 11 11 −13 1 −2 −7 12} {−58 −20 27 −2 −27 75 −29 0 47 −42 −11 11 −9 −3 19 −4} {−51 23 −22 5−63 3 37 −5 1 64 −35 −4 29 −31 −11 13} {−27 −76 49 −2 40 14 9 −17 −56 36−25 6 14 3 −6 8} {19 −4 −36 22 52 7 36 −23 28 −17 −64 15 −5 −44 48 9}{29 50 13 −10 1 34 −59 1 −51 4 −16 30 52 −33 24 −5} {−12 −21 −74 43 −1339 18 −5 −58 −35 27 −5 19 26 6 −5} {19 38 −10 −5 28 66 0 −5 −4 19 −30−26 −40 28 −60 37} [[{−6 27 18 −5 −37 −18 12 −25 −44 −10 −38 37 −66 4540 −7} {−13 −28 −45 −39 0 −5 −39 69 −23 16 −12 −18 −50 −31 24 13} {−1 824 −51 −15 −9 44 10 −28 −70 −12 −39 24 −18 −4 51} {−8 −22 −17 33 −18 −45−57 −27 0 −31 −30 29 −2 −13 −53 49} {1 12 32 51 −8 8 −2 −31 −22 4 46 −39−49 −67 14 17} {4 5 24 60 −5 −14 −23 38 9 8 −34 −59 24 47 42 28} {−1 −5−20 −34 4 4 −15 −46 18 31 42 10 10 27 49 78} {−3 −7 −22 −34 −5 −11 −36−69 −1 −3 −25 −73 5 4 4 −49}]]},

-   -   Otherwise, if nTrS is equal to 16, lfnstTrSetIdx is equal to 2,        and lfnstIdx is equal to 1, the following applies:

lowFreqTransMatrix[m][n]= {{−112 47 −2 2 −34 13 2 0 15 −7 1 0 8 −3 −1 0}{29 −7 1 −1 −108 40 2 0 −45 13 4 −1 8 −5 1 0} {−36 −87 69 −10 −17 −33 26−2 7 14 −11 2 6 8 −7 0} {28 −5 2 −2 −29 13 −2 0 103 −36 −4 1 48 −16 −41} {−12 −24 15 −3 26 80 −61 9 15 54 −36 2 0 −4 6 −2} {18 53 69 −74 14 2428 −30 −6 −7 −11 12 −5 −7 −6 8} {5 −1 2 0 −26 6 0 1 45 −9 −1 0 −113 28 8−1} {−13 −32 18 −2 15 34 −27 7 −25 −80 47 −1 −16 −50 28 2} [[{−4 −13 −1019 18 46 60 −48 16 33 60 −48 1 0 5 −2} {15 33 63 89 8 15 25 40 −4 −8 −15−8 −2 −6 −9 −7} {−8 −24 −27 15 12 41 26 −29 −17 −50 −39 27 0 35 −67 26}{−2 −6 −24 13 −1 −8 37 −22 3 18 −51 22 −23 −95 17 17} {−3 −7 −16 −21 1024 46 75 8 20 38 72 1 2 1 7} {2 6 10 −3 −5 −16 −31 12 7 24 41 −16 −16−41 −89 49} {4 8 21 40 −4 −11 −28 −57 5 14 31 70 7 18 32 52} {0 1 4 11−2 −4 −13 −34 3 7 20 47 −6 −19 −42 −101}]]},

-   -   Otherwise, if nTrS is equal to 16, lfnstTrSetIdx is equal to 2,        and lfnstIdx is equal to 2, the following applies:

lowFreqTransMatrix[m][n]= {{−99 39 −1 2 65 −20 −5 0 −15 −2 5 −1 0 3 −10} {58 42 −33 3 33 −63 23 −1 −55 32 3 −5 21 −2 −8 3} {−15 71 −44 5 −58−29 25 3 62 −7 −4 −4 −19 4 0 1} {46 5 4 −6 71 −12 −15 5 52 −38 13 −2 −6323 3 −3} {−14 −54 −29 29 25 −9 61 −29 27 44 −48 5 −27 −21 12 7} {−3 3 69−42 −11 −50 −26 26 24 63 −19 −5 −18 −22 12 0} {17 16 −2 1 38 18 −12 0 621 −14 5 89 −42 8 −2} {15 54 −8 6 6 60 −26 −8 −30 17 −38 22 −43 −45 42−7} [[{−6 −17 −55 −28 9 30 −8 58 4 34 41 −52 −16 −36 −20 16} {−2 −1 −9−79 7 11 48 44 −13 −34 −55 6 12 23 20 −11} {7 29 14 −6 12 53 10 −11 1459 −15 −3 5 71 −54 13} {−5 −24 −53 15 −3 −15 −61 26 6 30 −16 23 13 56 44−35} {4 8 21 52 −1 −1 −5 29 −7 −17 −44 −84 8 20 31 39} {−2 −11 −25 −4 −4−21 −53 2 −5 −26 −64 19 −8 −19 −73 39} {−3 −5 −23 −57 −2 −4 −24 −75 1 39 −25 6 15 41 61} {1 1 7 18 1 2 16 47 2 5 24 67 3 9 25 88}]]},

-   -   Otherwise, if nTrS is equal to 16, lfnstTrSetIdx is equal to 3,        and lfnstIdx is equal to 1, the following applies:

lowFreqTransMatrix[m][n]= {{−114 37 3 2 −22 −23 14 0 21 −17 −5 2 5 2 −4−1} {−19 −41 19 −2 85 −60 −11 7 17 31 −34 2 −11 19 2 −8} {36 −25 18 −2−42 −53 35 5 46 −60 −25 19 8 21 −33 −1} {−27 −80 44 −3 −58 1 −29 19 −4118 −12 −7 12 −17 7 −6} {−11 −21 37 −10 44 −4 47 −12 −37 −41 58 18 10 −46−16 31} {15 47 10 −6 −16 −44 42 10 −80 25 −40 21 −23 −2 3 −14} {13 25 79−39 −13 10 31 −4 49 45 12 −8 3 −1 43 7} {16 11 −26 13 −13 −74 −20 −1 5−6 29 −47 26 −49 54 2} [[{−8 −34 −26 7 −26 −19 29 −37 1 22 46 −9 −81 3714 20} {−6 −30 −42 −12 −3 5 57 −52 −2 37 −12 6 74 10 6 −15} {5 9 −6 42−15 −18 −9 26 15 58 14 43 23 −10 −37 75} {−5 −23 −23 36 3 22 36 40 27 −4−16 56 −25 −46 56 −24} {1 3 23 73 8 5 34 46 −12 2 35 −38 26 52 2 −31}{−3 −2 −21 −52 1 −10 −17 44 −19 −20 30 45 27 61 49 21} {−2 −7 −33 −56 −4−6 21 63 15 31 32 −22 −10 −26 −52 −38} {−5 −12 −18 −12 8 22 38 36 −5 −15−51 −63 −5 0 15 73}]]},

-   -   Otherwise, if nTrS is equal to 16, lfnstTrSetIdx is equal to 3,        and lfnstIdx is equal to 2, the following applies:

lowFreqTransMatrix[m][n]= {{−102 22 7 2 66 −25 −6 −1 −15 14 1 −1 2 −2 10} {12 93 −27 −6 −27 −64 36 6 13 5 −23 0 −2 6 5 −3} {−59 −24 17 1 −62 −2−3 2 83 −12 −17 −2 −24 14 7 −2} {−33 23 −36 11 −21 50 35 −16 −23 −78 1619 22 15 −30 −5} {0 −38 −81 30 27 5 51 −32 24 36 −16 12 −24 −8 9 1} {2838 8 −9 62 32 −13 2 51 −32 15 5 −66 28 0 −1} {11 −35 21 −17 30 −18 31 18−11 −36 −80 12 16 49 13 −32} {−13 23 22 −36 −12 64 39 25 −19 23 −36 9−30 −58 33 −7} [[{−9 −20 −55 −83 3 −2 1 62 8 2 27 −28 7 15 −11 5} {−6 24−38 23 −8 40 −49 0 −7 9 −25 −44 23 39 70 −3} {12 17 17 0 32 27 21 2 6711 −6 −10 89 −22 −12 16} {2 −9 8 45 7 −8 27 35 −9 −31 −17 −87 −23 −22−19 44} {−1 −9 28 −24 −1 −10 49 −30 −8 −7 40 1 4 33 65 67} {5 −12 −24−17 13 −34 −32 −16 14 −67 −7 9 7 −74 49 1} {2 −6 11 45 3 −10 33 55 8 −559 4 7 −4 44 −66} {−1 1 −14 36 −1 2 −20 69 0 0 −15 72 3 4 5 65}]]},

-   -   Otherwise, if nTrS is equal to 48, lfnstTrSetIdx is equal to 0,        and lfnstIdx is equal to 1 the following applies:

The zero-out range is unified for all blocks. The newly added parts ontop of JVET-02001 are highlighted in bold and Italic and the deletedparts are highlighted in are marked with double brackets (e.g., [[a]]denotes the deletion of the character “a”).

7.3.8.12 Residual Coding Syntax

Descriptor residual_coding( x0, y0, log2TbWidth, log2TbHeight, cIdx ) {...  if( lastSubBlock = = 0 && log2TbWidth >= 2 && log2TbHeight >= 2 &&  !transform_skip_flag[ x0 ][ y0 ] && lastScanPos > 0 )   LfnstDcOnly =0  if([[ ( ]] lastSubBlock > 0 [[&& log2TbWidth >= 2 && log2TbHeight >=2 ) ]] ||   [[ (]] lastScanPos > 7 [[&& ( log2TbWidth = = 2 ||log2TbWidth = = 3 ) &&   log2TbWidth = = log2TbHeight )]] )  LfnstZeroOutSigCoeffFlag = 0  QState = 0 ...8.7.4.2 GeneralInputs to this process are:

-   -   a luma location (xTbY, yTbY) specifying the top-left sample of        the current luma transform block relative to the top-left luma        sample of the current picture,    -   a variable nTbW specifying the width of the current transform        block,    -   a variable nTbH specifying the height of the current transform        block,    -   a variable cIdx specifying the colour component of the current        block,    -   an (nTbW)×(nTbH) array d[x] [y] of scaled transform coefficients        with x=0 . . . nTbW−1, y=0 . . . nTbH−1.        Output of this process is the (nTbW)×(nTbH) array r[x] [y] of        residual samples with x=0 . . . nTbW−1, y=0 . . . nTbH−1.        When lfnst_idx[xTbY] [yTbY] is not equal to 0 and both nTbW and        nTbH are greater than or equal to 4, the following applies:    -   The variables predModeIntra, nLfnstOutSize, log 2LfnstSize,        nLfnstSize, and nonZeroSize are derived as follows:        predModeIntra=(cIdx==0)?IntraPredModeY[xTbY][yTbY]:IntraPredModeC[xTbY][yTbY]  (8-965)        nLfnstOutSize=(nTbW>=8&& nTbH>=8)?48:16  (8-966)        log 2LfnstSize=(nTbW>=8&& nTbH>=8)?3:2  (8-967)        nLfnstSize=1<<log 2LfnstSize  (8-968)        nonZeroSize[[=((nTbW==4&& nTbH==4)∥(nTbW==8&&        nTbH==8))?]]8[[:16]]  (8-969)        When intra_mip_flag[xTbComp] [yTbComp] is equal to 1 and cIdx is        equal to 0, predModeIntra is set equal to INTRA_PLANAR.

This embodiment gives an example for 16×32 residual pattern applied toM×N (M>=8, N>=8) blocks coded with RST (a.k.a LFNST). Alternatively,this embodiment gives an example for blocks coded with RST (a.k.a LFNST)and nLfnstSize is equal to 8.

The newly added parts on top of JVET-02001 are highlighted in bold andItalic and the deleted parts are highlighted in are marked with doublebrackets (e.g., [[a]] denotes the deletion of the character “a”).

Transformation Process for Scaled Transform Coefficients

8.7.4.1 General

Inputs to this process are:

-   -   a luma location (xTbY, yTbY) specifying the top-left sample of        the current luma transform block relative to the top-left luma        sample of the current picture,    -   a variable nTbW specifying the width of the current transform        block,    -   a variable nTbH specifying the height of the current transform        block,    -   a variable cIdx specifying the colour component of the current        block,    -   an (nTbW)×(nTbH) array d[x] [y] of scaled transform coefficients        with x=0 . . . nTbW−1, y=0 . . . nTbH−1.        Output of this process is the (nTbW)×(nTbH) array r[x] [y] of        residual samples with x=0 . . . nTbW−1, y=0 . . . nTbH−1.        When lfnst_idx[xTbY] [yTbY] is not equal to 0 and both nTbW and        nTbH are greater than or equal to 4, the following applies:    -   The variables predModeIntra, nLfnstOutSize, log 2LfnstSize,        nLfnstSize, and nonZeroSize are derived as follows:        predModeIntra=(cIdx==0)?IntraPredModeY[xTbY][yTbY]:IntraPredModeC[xTbY][yTbY]  (8-965)        nLfnstOutSize=(nTbW>=8&& nTbH>=8)?[[        36:16  (8-966)        log 2LfnstSize=(nTbW>=8&& nTbH>=8)?3:2  (8-967)        nLfnstSize=1<<log 2LfnstSize  (8-968)        nonZeroSize=((nTbW==4&& nTbH==4)∥(nTbW==8&&        nTbH==8))?8:16  (8-969)    -   When intra_mip_flag[xTbComp] [yTbComp] is equal to 1 and cIdx is        equal to 0, predModeIntra is set equal to INTRA_PLANAR.    -   When predModeIntra is equal to either INTRA_LT_CCLM,        INTRA_L_CCLM, or INTRA_T_CCLM, predModeIntra is set equal to        IntraPredModeY[xTbY+nTbW/2] [yTbY+nTbH/2].    -   The wide angle intra prediction mode mapping process as        specified in clause 8.4.5.2.6 is invoked with predModeIntra,        nTbW, nTbH and cIdx as inputs, and the modified predModeIntra as        output.    -   The values of the list u[x] with x=0 . . . nonZeroSize−1 are        derived as follows:        xC=DiagScanOrder[2][2][x][0]   (8-970)        yC=DiagScanOrder[2][2][x][1]   (8-971)        u[x]=d[xC][yC]  (8-972)    -   The one-dimensional low frequency non-separable transformation        process as specified in clause 8.7.4.2 is invoked with the input        length of the scaled transform coefficients nonZeroSize, the        transform output length nTrS set equal to nLfnstOutSize, the        list of scaled non-zero transform coefficients u[x] with x=0 . .        . nonZeroSize−1, the intra prediction mode for LFNST set        selection predModeIntra, and the LFNST index for transform        selection in the selected LFNST set lfnst_idx[xTbY] [yTbY] as        inputs, and the list v[x] with x=0 . . . nLfnstOutSize−1 as        output.    -   The array d[x] [y] with x=0 . . . nLfnstSize−1, y=0 . . .        nLfnstSize−1 is derived as follows:        -   -   If predModeIntra is less than or equal to 34, the                following applies:                d[x][y]=(y<4)?v[x+(y<<log                2LfnstSize)]:((x<4)?v[32+x+((y−4)<<2)]:d[x][y])  (8-973)

        -   Otherwise, the following applies:            d[x][y]=(x<4)?v[y+(x<<log            2LfnstSize)]:((y<4)?v[32+y+((x−4)<<2)]:d[x][y])  (8-974)

        -   -   -   

                -   

            -   

            -   

            -   

            -                   The variable implicitMtsEnabled is derived as follows:    -   If sps_mts_enabled_flag is equal to 1 and one of the following        conditions is true, implicitMtsEnabled is set equal to 1:        -   IntraSubPartitionsSplitType is not equal to ISP_NO_SPLIT        -   cu_sbt_flag is equal to 1 and Max(nTbW, nTbH) is less than            or equal to 32        -   sps_explicit_mts_intra_enabled_flag is equal to 0 and            CuPredMode[0] [xTbY] [yTbY] is equal to MODE_INTRA and            lfnst_idx[x0] [y0] is equal to 0 and intra_mip_flag[x0] [y0]            is equal to 0    -   Otherwise, implicitMtsEnabled is set equal to 0.        8.7.4.3 Low Frequency Non-Separable Transformation Matrix        Derivation Process        Inputs to this process are:    -   a variable nTrS specifying the transform output length,    -   a variable predModeIntra specifying the intra prediction mode        for LFNST set selection,    -   a variable lfnstIdx specifying the LFNST index for transform        selection in the selected LFNST set.        Output of this process is the transformation matrix        lowFreqTransMatrix.        The variable lfnstTrSetIdx is specified in Table 8-16 depending        on predModeIntra.

TABLE 8-16 Specification of lfnstTrSetIdx predModeIntra lfnstTrSetIdxpredModeIntra < 0 1 0 <= predModeIntra <= 1 0  2 <= predModeIntra <= 121 13 <= predModeIntra <= 23 2 24 <= predModeIntra <= 44 3 45 <=predModeIntra <= 55 2 56 <= predModeIntra <= 80 1The transformation matrix lowFreqTransMatrix is derived based on nTrS,lfnstTrSetIdx, and lfnstIdx as follows:

-   -   If nTrS is equal to 16, lfnstTrSetIdx is equal to 0, and        lfnstIdx is equal to 1, the following applies:

lowFreqTransMatrix[m][n]= {{108 −44 −15 1 −44 19 7 −1 −11 6 2 −1 0 −1 −10} {−40 −97 56 12 −11 29 −12 −3 18 18 −15 −3 −1 −3 2 1} {25 −31 −1 7 100−16 −29 1 −54 21 14 −4 −7 2 4 0} {−32 −39 −92 51 −6 −16 36 −8 3 22 18−15 4 1 −5 2} {8 −9 33 −8 −16 −102 36 23 −4 38 −27 −5 5 16 −8 −6} {−25 516 −3 −38 14 11 −3 −97 7 26 1 55 −10 −19 3} {8 9 16 1 37 36 94 −38 −7 3−47 11 −6 −13 −17 10} {2 34 −5 1 −7 24 −25 −3 8 99 −28 −29 6 −43 21 11}{−16 −27 −39 −109 6 10 16 24 3 19 10 24 −4 −7 −2 −3} {−9 −10 −34 4 −9 −5−29 5 −33 −26 −96 33 14 4 39 −14} {−13 1 4 −9 −30 −17 −3 −64 −35 11 1719 −86 6 36 14} {8 −7 −5 −15 7 −30 −28 −87 31 4 4 33 61 −5 −17 22} {−213 −6 −4 −2 28 −13 −14 −3 37 −15 −3 −2 107 −36 −24} {4 9 11 31 4 9 16 1912 33 32 94 12 0 34 −45} {2 −2 8 −16 8 5 28 −17 6 −7 18 −45 40 36 97 −8}{0 −2 0 −10 −1 −7 −3 −35 −1 −7 −2 −32 −6 −33 −16 −112}},

-   -   Otherwise, if nTrS is equal to 16, lfnstTrSetIdx is equal to 0,        and lfnstIdx is equal to 2, the following applies:

lowFreqTransMatrix[m][n]= {{119 −30 −22 −3 −23 −2 3 2 −16 3 6 0 −3 2 10} {−27 −101 31 17 −47 2 22 3 19 30 −7 −9 5 3 −5 −1} {0 58 22 −15 −102 238 2 10 −13 −5 4 14 −1 −9 0} {23 4 66 −11 22 89 −2 −26 13 −8 −38 −1 −9−20 −2 8} {−19 −5 −89 2 −26 76 −11 −17 20 13 18 −4 1 −15 3 5} {−10 −1 −16 23 25 87 −7 −74 4 39 −5 0 −1 −20 −1} {−17 −28 12 −8 −32 14 −53 −6 −68−67 17 29 2 6 25 4} {1 −24 −23 1 17 −7 52 9 50 −92 −15 27 −15 −10 −6 3}{−6 −17 −2 −111 7 −17 8 −42 9 18 16 25 −4 2 −1 11} {9 5 35 0 6 21 −9 3444 −3 102 11 −7 13 11 −20} {4 −5 −5 −10 15 19 −2 6 6 −12 −13 6 95 69 −29−24} {−6 −4 −9 −39 1 22 0 102 −19 19 −32 30 −16 −14 −8 −23} {4 −4 7 8 4−13 −18 5 0 0 21 22 58 −88 −54 28} {−4 −7 0 −24 −7 0 −25 3 −3 −30 8 −76−34 4 −80 −26} {0 6 0 30 −6 1 −13 −23 1 20 −2 80 −44 37 −68 1} {0 0 −1 5−1 −7 1 −34 −2 3 −6 19 5 −38 11 −115}},

-   -   Otherwise, if nTrS is equal to 16, lfnstTrSetIdx is equal to 1,        and lfnstIdx is equal to 1, the following applies:

lowFreqTransMatrix[m][n]= {{−111 39 4 3 44 11 −12 −1 7 −16 −5 2 3 −1 42} {−47 −27 15 −1 −92 43 20 −2 20 39 −16 −5 10 −5 −13 2} {−35 −23 4 4−17 −72 32 6 −59 18 50 −6 0 40 0 −13} {13 93 −27 −4 −48 13 −34 4 −52 111 10 3 16 −3 1} {−11 −27 1 2 −47 −4 −36 10 −2 −85 14 29 −20 −2 57 4} {0−35 32 −2 26 60 −3 −17 −82 1 −30 0 −37 21 3 12} {−17 −46 −92 14 7 −10−39 29 −17 27 −28 17 1 −15 −13 17} {4 −10 −23 4 16 58 −17 26 30 21 67 2−13 59 13 −40} {5 −20 32 −5 8 −3 −46 −7 −4 2 −15 24 100 44 0 5} {−4 −138 −18 −7 −42 −63 −6 33 34 −23 15 −65 33 −20 2} {−2 −10 35 −19 5 8 −4414 −25 25 58 17 7 −84 −16 −18} {5 13 18 34 11 −4 18 18 5 58 −3 42 −2 −1085 38} {−5 −7 −34 −83 2 −1 −4 −73 4 20 15 −12 4 −3 44 12} {0 4 −2 −60 59 42 34 5 −14 9 80 −5 13 −38 37} {−1 2 7 −57 3 −7 9 68 −9 6 −49 −20 6 −436 −64} {−1 0 −12 23 1 −4 17 −53 −3 4 −21 72 −4 −8 −3 −83}},

-   -   Otherwise, if nTrS is equal to 16, lfnstTrSetIdx is equal to 1,        and lfnstIdx is equal to 2, the following applies:

lowFreqTransMatrix[m][n]= {{88 −55 6 −3 −66 27 9 −2 11 11 −13 1 −2 −7 12} {−58 −20 27 −2 −27 75 −29 0 47 −42 −11 11 −9 −3 19 −4} {−51 23 −22 5−63 3 37 −5 1 64 −35 −4 29 −31 −11 13} {−27 −76 49 −2 40 14 9 −17 −56 36−25 6 14 3 −6 8} {19 −4 −36 22 52 7 36 −23 28 −17 −64 15 −5 −44 48 9}{29 50 13 −10 1 34 −59 1 −51 4 −16 30 52 −33 24 −5} {−12 −21 −74 43 −1339 18 −5 −58 −35 27 −5 19 26 6 −5} {19 38 −10 −5 28 66 0 −5 −4 19 −30−26 −40 28 −60 37} {−6 27 18 −5 −37 −18 12 −25 −44 −10 −38 37 −66 45 40−7} {−13 −28 −45 −39 0 −5 −39 69 −23 16 −12 −18 −50 −31 24 13} {−1 8 24−51 −15 −9 44 10 −28 −70 −12 −39 24 −18 −4 51} {−8 −22 −17 33 −18 −45−57 −27 0 −31 −30 29 −2 −13 −53 49} {1 12 32 51 −8 8 −2 −31 −22 4 46 −39−49 −67 14 17} {4 5 24 60 −5 −14 −23 38 9 8 −34 −59 24 47 42 28} {−1 −5−20 −34 4 4 −15 −46 18 31 42 10 10 27 49 78} {−3 −7 −22 −34 −5 −11 −36−69 −1 −3 −25 −73 5 4 4 −49}},

-   -   Otherwise, if nTrS is equal to 16, lfnstTrSetIdx is equal to 2,        and lfnstIdx is equal to 1, the following applies:

lowFreqTransMatrix[m][n]= {{−112 47 −2 2 −34 13 2 0 15 −7 1 0 8 −3 −1 0}{29 −7 1 −1 −108 40 2 0 −45 13 4 −1 8 −5 1 0} {−36 −87 69 −10 −17 −33 26−2 7 14 −11 2 6 8 −7 0} {28 −5 2 −2 −29 13 −2 0 103 −36 −4 1 48 −16 −41} {−12 −24 15 −3 26 80 −61 9 15 54 −36 2 0 −4 6 −2} {18 53 69 −74 14 2428 −30 −6 −7 −11 12 −5 −7 −6 8} {5 −1 2 0 −26 6 0 1 45 −9 −1 0 −113 28 8−1} {−13 −32 18 −2 15 34 −27 7 −25 −80 47 −1 −16 −50 28 2} {−4 −13 −1019 18 46 60 −48 16 33 60 −48 1 0 5 −2} {15 33 63 89 8 15 25 40 −4 −8 −15−8 −2 −6 −9 −7} {−8 −24 −27 15 12 41 26 −29 −17 −50 −39 27 0 35 −67 26}{−2 −6 −24 13 −1 −8 37 −22 3 18 −51 22 −23 −95 17 17} {−3 −7 −16 −21 1024 46 75 8 20 38 72 1 2 1 7} {2 6 10 −3 −5 −16 −31 12 7 24 41 −16 −16−41 −89 49} {4 8 21 40 −4 −11 −28 −57 5 14 31 70 7 18 32 52} {0 1 4 11−2 −4 −13 −34 3 7 20 47 −6 −19 −42 −101}},

-   -   Otherwise, if nTrS is equal to 16, lfnstTrSetIdx is equal to 2,        and lfnstIdx is equal to 2, the following applies:

lowFreqTransMatrix[m][n]= {{−99 39 −1 2 65 −20 −5 0 −15 −2 5 −1 0 3 −10} {58 42 −33 3 33 −63 23 −1 −55 32 3 −5 21 −2 −8 3} {−15 71 −44 5 −58−29 25 3 62 −7 −4 −4 −19 4 0 1} {46 5 4 −6 71 −12 −15 5 52 −38 13 −2 −6323 3 −3} {−14 −54 −29 29 25 −9 61 −29 27 44 −48 5 −27 −21 12 7} {−3 3 69−42 −11 −50 −26 26 24 63 −19 −5 −18 −22 12 0} {17 16 −2 1 38 18 −12 0 621 −14 5 89 −42 8 −2} {15 54 −8 6 6 60 −26 −8 −30 17 −38 22 −43 −45 42−7} {−6 −17 −55 −28 9 30 −8 58 4 34 41 −52 −16 −36 −20 16} {−2 −1 −9 −797 11 48 44 −13 −34 −55 6 12 23 20 −11} {7 29 14 −6 12 53 10 −11 14 59−15 −3 5 71 −54 13} {−5 −24 −53 15 −3 −15 −61 26 6 30 −16 23 13 56 44−35} {4 8 21 52 −1 −1 −5 29 −7 −17 −44 −84 8 20 31 39} {−2 −11 −25 −4 −4−21 −53 2 −5 −26 −64 19 −8 −19 −73 39} {−3 −5 −23 −57 −2 −4 −24 −75 1 39 −25 6 15 41 61} {1 1 7 18 1 2 16 47 2 5 24 67 3 9 25 88}},

-   -   Otherwise, if nTrS is equal to 16, lfnstTrSetIdx is equal to 3,        and lfnstIdx is equal to 1, the following applies:

lowFreqTransMatrix[m][n]= {{−114 37 3 2 −22 −23 14 0 21 −17 −5 2 5 2 −4−1} {−19 −41 19 −2 85 −60 −11 7 17 31 −34 2 −11 19 2 −8} {36 −25 18 −2−42 −53 35 5 46 −60 −25 19 8 21 −33 −1} {−27 −80 44 −3 −58 1 −29 19 −4118 −12 −7 12 −17 7 −6} {−11 −21 37 −10 44 −4 47 −12 −37 −41 58 18 10 −46−16 31} {15 47 10 −6 −16 −44 42 10 −80 25 −40 21 −23 −2 3 −14} {13 25 79−39 −13 10 31 −4 49 45 12 −8 3 −1 43 7} {16 11 −26 13 −13 −74 −20 −1 5−6 29 −47 26 −49 54 2} {−8 −34 −26 7 −26 −19 29 −37 1 22 46 −9 −81 37 1420} {−6 −30 −42 −12 −3 5 57 −52 −2 37 −12 6 74 10 6 −15} {5 9 −6 42 −15−18 −9 26 15 58 14 43 23 −10 −37 75} {−5 −23 −23 36 3 22 36 40 27 −4 −1656 −25 −46 56 −24} {1 3 23 73 8 5 34 46 −12 2 35 −38 26 52 2 −31} {−3 −2−21 −52 1 −10 −17 44 −19 −20 30 45 27 61 49 21} {−2 −7 −33 −56 −4 −6 2163 15 31 32 −22 −10 −26 −52 −38} {−5 −12 −18 −12 8 22 38 36 −5 −15 −51−63 −5 0 15 73}},

-   -   Otherwise, if nTrS is equal to 16, lfnstTrSetIdx is equal to 3,        and lfnstIdx is equal to 2, the following applies:

lowFreqTransMatrix[m][n]= {{−102 22 7 2 66 −25 −6 −1 −15 14 1 −1 2 −2 10} {12 93 −27 −6 −27 −64 36 6 13 5 −23 0 −2 6 5 −3} {−59 −24 17 1 −62 −2−3 2 83 −12 −17 −2 −24 14 7 −2} {−33 23 −36 11 −21 50 35 −16 −23 −78 1619 22 15 −30 −5} {0 −38 −81 30 27 5 51 −32 24 36 −16 12 −24 −8 9 1} {2838 8 −9 62 32 −13 2 51 −32 15 5 −66 28 0 −1} {11 −35 21 −17 30 −18 31 18−11 −36 −80 12 16 49 13 −32} {−13 23 22 −36 −12 64 39 25 −19 23 −36 9−30 −58 33 −7} {−9 −20 −55 −83 3 −2 1 62 8 2 27 −28 7 15 −11 5} {−6 24−38 23 −8 40 −49 0 −7 9 −25 −44 23 39 70 −3} {12 17 17 0 32 27 21 2 6711 −6 −10 89 −22 −12 16} {2 −9 8 45 7 −8 27 35 −9 −31 −17 −87 −23 −22−19 44} {−1 −9 28 −24 −1 −10 49 −30 −8 −7 40 1 4 33 65 67} {5 −12 −24−17 13 −34 −32 −16 14 −67 −7 9 7 −74 49 1} {2 −6 11 45 3 −10 33 55 8 −559 4 7 −4 44 −66} {−1 1 −14 36 −1 2 −20 69 0 0 −15 72 3 4 5 65}},

-   -   Otherwise, if nTrS is equal to [[48]]        , lfnstTrSetIdx is equal to 0, and lfnstIdx is equal to 1 the        following applies:        -   lowFreqTransMatrix[m] [n]=lowFreqTransMatrixCol0to15[m] [n]            with m=0 . . . [[15]]            , n=0 . . . 15

lowFreqTransMatrixCol0to15= {{−117 28 18 2 4 1 [[2 1]] 32 −18 −2 0 −1 0[[0 0]]} {−29 −91 47 1 9 0 [[3 0]] −54 26 −8 3 0 1 [[0 0]]} {−10 62 −11−8 −2 −2 [[−1 −1]] −95 3 32 0 4 0 [[2 0]]} {−15 15 −10 −2 1 0 [[1 0]] 10112 −20 −17 −4 −4 [[−1 −2]]} {32 39 92 −44 4 −10 [[1 −4]] 26 12 −15 13−5 2 [[−2 0]]} {−10 1 50 −15 2 −3 [[1 −1]] −28 −15 14 6 1 1 [[1 0]]} {1−33 −11 −14 7 −2 [[2 0]] 29 −12 37 −7 −4 0 [[−1 0]] } {0 6 −6 21 −4 2[[0 0]] −20 −24 −104 30 5 5 [[1 2]]} {−13 −13 −37 −101 29 −11 [[8 −3]]−12 −15 −20 2 −11 5 [[−2 1]]} {6 1 −14 −36 9 −3 [[2 0]] 10 9 −18 −1 −3 1[[0 0]]} {−12 −2 −26 −12 −9 2 [[−1 1]] −3 30 4 34 −4 0 [[−1 0]]} {0 −3 0−4 −15 6 [[−3 1]] −7 −15 −28 −86 19 −5 [[4 −1]]} {−1 9 13 5 14 −2 [[2−1]] −8 3 −4 −62 4 1 [[1 0]]} {6 2 −3 2 10 −1 [[2 0]] 8 3 −1 −20 0 1 [[00]]} {6 9 −2 35 110 −22 [[11 −4]] −2 0 −3 1 −18 12 [[−3 2]]} {−1 7 −2 9−11 5 [[−1 1]] −7 2 −22 4 −13 0 [[−1 0]]}},

-   -   -   lowFreqTransMatrix[m]            [n]=lowFreqTransMatrixCol16to31[m−[[16]]            ] [n] with m=[[16]]            . . . [[31]]            , n=0 . . . 15

lowFreqTransMatrixCol16to31= { { 14 −1 −3 0 −1 0 [[ 0 0 ]] 2 0 0 0 0 0[[ 0 0 ]] } { 33 5 −9 −1 −2 0 [[ −1 0 ]] −3 3 0 0 0 0 [[ 0 0 ]] } { 32−30 −4 4 −1 1 [[ 0 0 ]] 6 2 −5 0 0 0 [[ 0 0 ]] } { −20 −26 31 1 0 0 [[ 00 ]] 2 −16 −1 6 0 1 [[ 0 0 ]] } { 29 −16 −22 8 0 1 [[ 0 1 ]] −20 6 4 −31 0 [[ 0 0 ]] } { −99 −4 9 5 5 2 [[ 2 1 ]] 44 −10 −11 1 −2 0 [[ −1 0 ]]} { 6 −99 3 26 −1 5 [[ 0 2 ]] 14 30 −27 −2 1 −1 [[ 0 −1 ]] } { −7 −46 10−14 7 0 [[ 1 0 ]] 9 21 7 −6 −2 −1 [[ 0 −1 ]] } { −12 10 26 12 −6 0 [[ −10 ]] −32 −2 11 3 3 −1 [[ 1 0 ]] } { 38 26 −13 −1 −5 −1 [[ −1 0 ]] 102 3−14 −1 −5 −1 [[ −2 0 ]] } { −30 3 −92 14 19 0 [[ 3 0 ]] −11 34 21 −33 1−2 [[ 0 −1 ]] } { −5 −17 −41 42 −6 2 [[ −1 1 ]] −1 −40 37 13 −4 2 [[ −11 ]] } { −12 23 16 −11 −17 0 [[ −1 0 ]] −11 97 −3 −3 0 −6 [[ 0 −2 ]] } {−4 4 −16 0 −2 0 [[ 1 0 ]] 34 23 6 −7 −4 −2 [[ −1 0 ]] } { −5 −4 −22 8−25 3 [[ 0 0 ]] −3 −21 2 −3 9 −2 [[ 1 0 ]] } { 0 28 0 76 4 −6 [[ 0 −2 ]]−13 5 −76 −4 33 −1 [[ 3 0 ]] } },

-   -   -   lowFreqTransMatrix[m]            [n]=lowFreqTransMatrixCol32to47[m−[[32]]            ] [n] with m=[[32]]            . . . [[47]]            , n=0 . . . 15

lowFreqTransMatrixCol32to47= {{3 0 −1 0 1 0 0 0 1 0 0 0 [[1 0 0 0]]} {72 −2 0 −1 1 0 0 2 1 −1 0 [[0 0 0 0]]} {6 −3 0 0 2 0 −1 0 2 −1 0 0 [[1 00 0]]} {1 −4 0 0 0 −3 0 1 0 −1 0 0 [[0 −2 0 0]]} {1 −4 −3 2 −4 1 0 0 1−1 −2 1 [[−2 0 0 0]]} {−5 4 −3 0 8 −1 −2 0 −2 1 −1 0 [[4 0 −1 0]]} {−6 66 −3 1 3 −3 0 −1 1 1 0 [[0 1 −1 0]]} {2 2 5 −2 0 3 4 −1 0 0 1 0 [[0 1 2−1]]} {11 −5 −1 6 −4 2 1 0 3 −1 1 2 [[−1 0 0 0]]} {−29 10 10 0 10 −4 −11 −7 1 2 1 [[2 −1 0 0]]} {−9 −4 18 3 2 0 0 −2 −1 −1 3 0 [[0 0 0 −1]]}{−10 13 −1 −4 4 −4 3 4 −2 2 −1 −1 [[1 −1 1 2]]} {−21 −5 23 0 2 −2 −1 6−3 −3 1 0 [[0 0 0 2]]} {108 −5 −30 6 −27 10 7 −2 11 −3 −1 1 [[−4 1 01]]} {−7 1 3 −5 3 0 −1 0 0 1 0 −1 [[1 0 0 0]]} {9 18 −3 −35 −4 −1 6 1 12 0 −3 [[−1 0 2 0]]}},

-   -   Otherwise, if nTrS is equal to [[48]]        , lfnstTrSetIdx is equal to 0, and lfnstIdx is equal to 2 the        following applies:        -   lowFreqTransMatrix[m] [n]=lowFreqTransMatrixCol0to15[m] [n]            with m=0 . . . [[15]]            , n=0 . . . 15

lowFreqTransMatrixCol0to15= { { −108 48 9 1 1 1 [[ 0 0 ]] 44 −6 −9 −1 −10 [[ −1 0 ]] } { 55 66 −37 −5 −6 −1 [[ −2 0 ]] 67 −30 −20 4 −2 0 [[ −1 0]] } { 2 86 −21 −13 −4 −2 [[ −1 −1 ]] −88 5 6 4 5 1 [[ 1 0 ]] } { −24−21 −38 19 0 4 [[ −1 2 ]] −23 −89 31 20 2 3 [[ 1 1 ]] } { 9 20 98 −26 −3−5 [[ 0 −2 ]] −9 −26 15 −16 2 0 [[ 1 0 ]] } { −21 −7 −37 10 2 2 [[ −1 1]] −10 69 −5 −7 −2 −2 [[ 0 −1 ]] } { −10 −25 4 −17 8 −2 [[ 2 −1 ]] −27−17 −71 25 8 2 [[ 1 1 ]] } { 2 5 10 64 −9 4 [[ −3 1 ]] −4 8 62 3 −17 1[[ −2 0 ]] } { −11 −15 −28 −97 6 −1 [[ 4 −1 ]] 7 3 57 −15 10 −2 [[ 0 −1]] } { 9 13 24 −6 7 −2 [[ 1 −1 ]] 16 39 20 47 −2 −2 [[ −2 0 ]] } { −7 1112 7 2 −1 [[ 0 −1 ]] −14 −1 −24 11 2 0 [[ 0 0 ]] } { 0 0 7 −6 23 −3 [[ 3−1 ]] 5 1 18 96 13 −9 [[ −1 −1 ]] } { −2 −6 −1 −10 0 1 [[ 1 0 ]] −7 −2−28 20 −15 4 [[ −3 1 ]] } { −1 6 −16 0 24 −3 [[ 1 −1 ]] 2 6 6 16 18 −7[[ 1 −1 ]] } { −5 −6 −3 −19 −104 18 [[ −4 3 ]] 0 6 0 35 −41 20 [[ −2 2]] } { −1 −2 0 23 −9 0 [[ −2 0 ]] 1 1 8 −1 29 1 [[ 1 0 ]] } },

-   -   -   lowFreqTransMatrix[m]            [n]=lowFreqTransMatrixCol16to31[m−[[16]]            ] [n] with m=[[16]]            . . . [[31]]            , n=0 . . . 15

lowFreqTransMatrixCol16to31= { { 9 −9 −1 1 0 0 [[ 0 0 ]] 3 −1 1 0 0 0 [[0 0 ]] } { −31 −19 14 4 1 1 [[ 1 0 ]] −6 3 5 −2 0 0 [[ 0 0 ]] } { 14 −50 3 0 0 [[ 0 0 ]] 10 −5 −2 0 −1 0 [[ 0 0 ]] } { −30 26 36 −8 −2 −2 [[ 0−1 ]] 14 18 −7 −9 −1 −1 [[ 0 0 ]] } { −61 −3 −2 3 7 1 [[ 1 0 ]] 12 16 −6−1 0 −1 [[ 0 0 ]] } { −93 2 19 0 3 0 [[ 2 0 ]] 17 4 0 0 −1 0 [[ 0 0 ]] }{ −4 −66 28 36 −5 3 [[ 0 1 ]] −10 20 33 −13 −8 0 [[ 0 −1 ]] } { −3 −75 5−14 1 4 [[ 0 1 ]] −36 3 18 −4 4 0 [[ 1 0 ]] } { −1 −27 13 6 1 −1 [[ 0 0]] −34 −6 0 3 4 1 [[ 2 0 ]] } { 28 23 76 −5 −25 −3 [[ −3 −1 ]] 6 36 −7−39 −4 −1 [[ 0 −1 ]] } { −20 48 11 −13 −5 −2 [[ 0 −1 ]] −105 −19 17 0 62 [[ 3 0 ]] } { −21 −7 −42 14 −24 −3 [[ 0 0 ]] 11 −47 −7 3 −5 9 [[ 1 2]] } { −2 −32 −2 −66 3 7 [[ 1 2 ]] −11 13 −70 5 43 −2 [[ 3 0 ]] } { −311 −63 9 4 −5 [[ 2 −1 ]] −22 94 −4 −6 −4 −4 [[ 1 −2 ]] } { −2 10 −18 1621 3 [[ −2 0 ]] −2 11 6 −10 6 −3 [[ −1 0 ]] } { 3 −6 13 76 30 −11 [[ −1−2 ]] −26 −8 −69 7 −9 −7 [[ 3 −1 ]] } },

-   -   -   lowFreqTransMatrix[m]            [n]=lowFreqTransMatrixCol32to47[m−[[32]]            ] [n] with m=[[32]]            . . . [[47]]            , n=0 . . . 15

lowFreqTransMatrixCol32to47= { { 1 −1 0 0 1 0 0 0 0 −1 0 0 [[ 0 0 0 0 ]]} { −7 −1 1 0 −1 1 1 0 −2 −1 1 0 [[ 0 0 0 0 ]] } { 6 −5 0 1 2 −1 0 0 1−1 0 0 [[ 1 0 0 0 ]] } { 1 3 −2 −1 3 2 −2 −1 0 1 0 0 [[ 1 1 −1 0 ]] } {2 0 −8 1 3 1 −1 1 0 −1 −2 0 [[ 1 0 −1 0 ]] } { 5 −4 −2 0 4 −2 0 1 0 0 00 [[ 2 −1 0 0 ]] } { 3 6 −3 −7 −1 3 3 −1 1 0 −1 0 [[ 0 1 1 −1 ]] } { 114 −2 −8 −2 1 −3 0 2 2 −1 −2 [[ 0 1 −1 0 ]] } { −2 8 1 5 −2 0 −3 1 1 1 02 [[ −1 0 −1 0 ]] } { 2 4 −18 −3 −1 −1 −2 −2 1 −2 −2 0 [[ 0 0 −1 −1 ]] }{ −14 8 8 2 1 2 −1 −2 3 0 −1 0 [[ 0 0 0 0 ]] } { 0 −1 19 −1 1 0 −1 −6 −11 2 0 [[ 1 0 0 −2 ]] } { 8 −14 −3 43 −1 2 7 −1 1 −2 1 3 [[ −1 1 1 0 ]] }{ 10 23 −19 −5 0 −6 −4 6 3 −2 1 1 [[ 0 −1 0 0 ]] } { −1 5 −1 −6 −1 −1 −1−1 −1 0 0 0 [[ 0 0 0 −1 ]] } { −10 −34 −25 13 −1 0 11 5 1 −1 1 −2 [[ 0 02 0 ]] } },

-   -   Otherwise, if nTrS is equal to [[48]]        , lfnstTrSetIdx is equal to 1, and lfnstIdx is equal to 1 the        following applies:        -   lowFreqTransMatrix[m] [n]=lowFreqTransMatrixCol0to15[m] [n]            with m=0 . . . [[15]]            , n=0 . . . 15

lowFreqTransMatrixCol0to15= { { 110 −49 −3 −4 −1 −1 [[ 0 −1 ]] −38 −1 100 2 0 [[ 1 0 ]] } { −43 −19 17 −1 3 0 [[ 1 0 ]] −98 46 14 −1 2 0 [[ 1 0]] } { −19 17 −7 3 −2 1 [[ −1 0 ]] −32 −59 29 3 4 0 [[ 2 0 ]] } { −35−103 39 1 7 0 [[ 2 0 ]] 38 −13 25 −6 1 −1 [[ 0 0 ]] } { 9 5 −6 −1 −1 0[[ −1 0 ]] 42 4 21 −11 1 −3 [[ 1 −1 ]] } { −5 −5 −28 9 −3 2 [[ −1 1 ]]−20 −78 22 16 1 3 [[ 0 1 ]] } { 14 17 27 −12 1 −3 [[ 1 −1 ]] 8 19 −13 4−2 1 [[ −1 0 ]] } { 7 35 17 −4 −1 0 [[ 0 0 ]] 3 8 54 −17 1 −2 [[ 1 −1 ]]} { −13 −27 −101 24 −8 6 [[ −3 2 ]] 11 43 6 28 −6 3 [[ −1 1 ]] } { −11−13 −3 −10 3 −1 [[ 1 0 ]] −19 −19 −37 8 4 2 [[ 0 1 ]] } { −4 −10 −24 −113 −2 [[ 0 −1 ]] −6 −37 −45 −17 8 −2 [[ 2 −1 ]] } { −2 1 13 −17 3 −5 [[ 1−2 ]] 3 0 −55 22 6 1 [[ 1 0 ]] } { 3 1 5 −15 1 −2 [[ 1 −1 ]] 7 4 −7 29−1 2 [[ −1 1 ]] } { −4 −8 −1 −50 6 −4 [[ 2 −2 ]] −1 5 −22 20 6 1 [[ 0 0]] } { 5 −1 26 102 −13 12 [[ −4 4 ]] −4 −2 −40 −7 −23 3 [[ −5 1 ]] } {−5 −6 −27 −22 −12 0 [[ −3 0 ]] −5 8 −20 −83 0 0 [[ 0 0 ]] } },

-   -   -   lowFreqTransMatrix[m]            [n]=lowFreqTransMatrixCol16to31[m−[[16]]            ] [n] with m=[[16]]            . . . [[31]]            , n=0 . . . 15

lowFreqTransMatrixCol16to31= { { −9 13 1 −2 0 0 [[ 0 0 ]] −4 2 −3 0 0 0[[ 0 0 ]] } { 26 26 −15 −3 −2 −1 [[ −1 0 ]] 11 −7 −9 2 0 0 [[ 0 0 ]] } {−72 43 34 −9 3 −2 [[ 1 −1 ]] 13 36 −18 −10 0 −2 [[ 0 −1 ]] } { −1 7 6 −71 −1 [[ 0 0 ]] −13 14 2 −4 2 −1 [[ 0 0 ]] } { 21 70 −32 −21 0 −4 [[ −1−1 ]] 34 −26 −57 11 4 2 [[ 0 1 ]] } { 80 −6 25 −5 −4 −1 [[ −1 0 ]] 6 −247 −9 0 0 [[ 0 0 ]] } { 48 −1 48 −15 −4 −2 [[ −1 −1 ]] 1 60 −28 −42 5 −6[[ 1 −2 ]] } { 10 14 −11 −34 4 −4 [[ 1 −1 ]] −80 −7 −6 2 15 0 [[ 3 0 ]]} { −3 14 21 −12 −7 −2 [[ −1 −1 ]] −23 10 −4 −12 3 0 [[ 1 0 ]] } { −12−30 3 −9 5 0 [[ 1 0 ]] −56 −9 −47 8 21 1 [[ 4 1 ]] } { 17 14 −58 14 15 0[[ 2 0 ]] −10 34 −7 28 4 −1 [[ 1 0 ]] } { 8 74 21 40 −14 0 [[ −2 0 ]]−36 −8 11 −13 −23 1 [[ −3 0 ]] } { 8 3 12 −14 −9 −1 [[ −1 0 ]] 4 29 −1531 10 4 [[ 1 1 ]] } { −16 −15 18 −29 −11 2 [[ −2 1 ]] 40 −45 −19 −22 312 [[ 4 1 ]] } { −1 5 8 −23 7 2 [[ 1 1 ]] 10 −11 −13 −3 12 −3 [[ 2 0 ]] }{ 9 7 24 −20 41 3 [[ 6 1 ]] 15 20 12 11 17 −9 [[ 1 −2 ]] } },

-   -   -   lowFreqTransMatrix[m]            [n]=lowFreqTransMatrixCol32to47[m−[[32]]            ] [n] with m=[[32]]            . . . [[47]]            , n=0 . . . 15

lowFreqTransMatrixCol32to47= { { −2 2 0 1 −1 1 0 0 −1 1 0 0 [[ −1 0 0 0]] } { 9 −3 −1 2 3 −3 0 0 4 −1 0 0 [[ 2 −1 0 0 ]] } { 3 0 −12 3 6 1 −3 21 −1 −2 0 [[ 3 1 −1 1 ]] } { −2 11 −6 −2 −2 4 −3 0 0 3 −2 0 [[ −1 1 −1 0]] } { −4 −32 5 24 1 −6 12 4 −3 −2 4 −2 [[ 0 −1 0 0 ]] } { −7 3 13 −4 −35 1 −5 −2 3 1 −2 [[ −1 2 −1 −2 ]] } { 11 −11 −51 11 −2 −10 −2 13 2 −6 −44 [[ −2 −3 2 2 ]] } { −16 46 1 3 2 7 −24 0 2 −2 −5 8 [[ 1 −1 −2 2 ]] } {2 9 −10 0 1 −5 −4 4 2 −2 2 2 [[ 0 −2 1 0 ]] } { −11 −30 10 59 −2 8 41 82 5 6 −7 [[ −1 3 5 −2 ]] } { 23 34 −31 4 10 −22 −30 22 4 −15 9 20 [[ 2−5 9 4 ]] } { −36 6 16 −14 2 19 −4 −12 −1 0 −7 −3 [[ 0 2 −2 −1 ]] } { 6122 55 14 13 3 −9 −65 1 −11 −21 −7 [[ 0 0 −1 3 ]] } { −25 41 0 12 9 7 −4212 −3 −14 2 28 [[ 5 1 6 2 ]] } { −9 23 4 9 14 9 −14 −4 0 −12 −7 6 [[ 3 06 3 ]] } { −26 −1 18 −1 −12 32 3 −18 −5 10 −25 −5 [[ −2 1 −8 10 ]] } },

-   -   Otherwise, if nTrS is equal to [[48]]        , lfnstTrSetIdx is equal to 1, and lfnstIdx is equal to 2 the        following applies:        -   lowFreqTransMatrix[m] [n]=lowFreqTransMatrixCol0to15[m] [n]            with m=0 . . . [[15]]            , n=0 . . . 15

lowFreqTransMatrixCol0to15= { { 80 −49 6 −4 1 −1 [[ 1 −1 ]] −72 36 4 0 10 [[ 0 0 ]] } { −72 −6 17 0 3 0 [[ 1 0 ]] −23 58 −21 2 −3 1 [[ −1 0 ]] }{ −50 19 −15 4 −1 1 [[ −1 1 ]] −58 −2 30 −3 4 −1 [[ 2 0 ]] } { −33 −4328 −7 4 −2 [[ 2 −1 ]] −38 11 −8 4 1 1 [[ 0 0 ]] } { 10 66 −21 −3 −3 0 [[−1 0 ]] −53 −41 −2 16 −1 4 [[ −1 1 ]] } { 18 14 13 −9 2 −2 [[ 1 −1 ]] 3432 −31 12 −5 2 [[ −2 1 ]] } { 21 66 −1 9 −4 2 [[ −1 1 ]] −21 41 −30 −100 −2 [[ 0 −1 ]] } { 1 −6 −24 17 −5 3 [[ −2 1 ]] 24 10 39 −21 5 −4 [[ 2−1 ]] } { 9 33 −24 1 4 0 [[ 1 0 ]] 6 50 26 1 −10 0 [[ −2 0 ]] } { −7 −9−32 14 −3 3 [[ −1 1 ]] −23 −28 0 −5 −1 0 [[ 0 0 ]] } { 6 30 69 −18 5 −4[[ 3 −1 ]] −3 −11 −34 −16 9 −4 [[ 2 −1 ]] } { 1 −8 24 −3 7 −2 [[ 2 −1 ]]−6 −51 −6 −4 −5 0 [[ −1 0 ]] } { 4 10 4 17 −9 4 [[ −2 1 ]] 5 14 32 −15 9−3 [[ 2 −1 ]] } { −3 −9 −23 10 −10 3 [[ −3 1 ]] −5 −14 −16 −27 13 −5 [[2 −1 ]] } { 2 11 22 2 9 −2 [[ 2 0 ]] −6 −7 20 −32 −3 −4 [[ 0 −1 ]] } { 2−3 8 14 −5 3 [[ −1 1 ]] −2 −11 5 −18 8 −3 [[ 2 −1 ]] } },

-   -   -   lowFreqTransMatrix[m]            [n]=lowFreqTransMatrixCol16to31[m−[[16]]            ] [n] with m=[[16]]            . . . [[31]]            , n=0 . . . 15

lowFreqTransMatrixCol16to31= { { 26 0 −12 2 −2 1 [[ −1 0 ]] −7 −9 6 1 00 [[ 0 0 ]] } { 55 −46 −1 6 −2 1 [[ −1 0 ]] −22 7 17 −7 2 −1 [[ 1 0 ]] }{ 6 57 −34 0 −2 0 [[ −1 0 ]] 34 −48 −2 14 −4 3 [[ −1 1 ]] } { −55 24 26−5 2 −1 [[ 1 0 ]] 15 46 −40 −1 −1 0 [[ −1 0 ]] } { 36 −5 41 −20 3 −3 [[1 −1 ]] −30 26 −32 −3 7 −2 [[ 2 −1 ]] } { 40 4 −4 −9 −3 −2 [[ −1 −1 ]]27 −31 −43 19 −2 3 [[ −1 1 ]] } { −35 −17 −3 26 −6 5 [[ −2 2 ]] 56 3 18−25 −1 −2 [[ −1 −1 ]] } { 33 32 −30 4 −3 −1 [[ −1 0 ]] −4 13 −16 −10 0−1 [[ 0 0 ]] } { −27 1 −28 −21 16 −5 [[ 3 −2 ]] −23 36 −2 40 −17 4 [[ −31 ]] } { −36 −59 −24 14 4 2 [[ 1 1 ]] −23 −26 23 26 −3 5 [[ 0 2 ]] } {−16 35 −35 30 −9 3 [[ −2 1 ]] −57 −13 6 4 −5 5 [[ −1 1 ]] } { 38 −1 0 256 2 [[ 1 1 ]] 47 20 35 1 −27 1 [[ −5 0 ]] } { 7 13 19 15 −8 1 [[ −1 0 ]]3 25 30 −18 1 −2 [[ 0 −1 ]] } { −1 −13 −30 11 −5 2 [[ −1 0 ]] −5 −8 −22−16 10 0 [[ 1 0 ]] } { 13 −5 −28 6 18 −4 [[ 3 −1 ]] −26 27 −14 6 −20 0[[ −2 0 ]] } { 12 −23 −19 22 2 0 [[ 1 0 ]] 23 41 −7 35 −10 4 [[ −1 1 ]]} },

-   -   -   lowFreqTransMatrix[m]            [n]=lowFreqTransMatrixCol32to47[m−[[32]]            ] [n] with m=[[32]]            . . . [[47]]            , n=0 . . . 15

lowFreqTransMatrixCol32to47= { { 3 5 1 2 2 2 1 1 1 1 0 0 [[ −1 −1 0 0 ]]} { 9 5 −12 1 −3 −4 4 2 4 1 −2 −1 [[ −1 −1 1 0 ]] } { −10 7 21 −10 6 1−11 0 −1 −1 4 2 [[ 3 0 −2 −1 ]] } { 17 −38 1 17 −3 11 15 −11 3 −1 −10 1[[ 0 1 3 2 ]] } { 15 −8 1 17 −1 −2 4 −8 2 0 −1 3 [[ 0 0 0 −1 ]] } { 7−49 52 10 −11 22 7 −26 −1 −6 −9 6 [[ −2 2 4 −2 ]] } { −15 −13 −27 9 9 −620 5 −3 2 −6 −9 [[ 3 −3 1 5 ]] } { 24 −26 −37 33 5 −32 55 −5 −7 22 −14−22 [[ 1 −9 −3 13 ]] } { 43 −13 4 −41 −19 −2 −24 17 11 −4 8 4 [[ −3 −3−3 −3 ]] } { 10 −26 38 7 −12 11 42 −22 −5 20 −14 −15 [[ −1 −2 1 6 ]] } {28 10 4 7 0 −15 7 −10 −1 7 −2 2 [[ 1 −3 0 0 ]] } { 37 −37 −9 −47 −28 5 018 8 6 0 −8 [[ −4 −3 −3 1 ]] } { 11 24 22 −11 −3 37 −13 −58 −5 12 −63 26[[ 9 −15 11 8 ]] } { 0 −29 −27 6 −27 −10 −30 9 −3 −10 −7 77 [[ 9 −13 45−8 ]] } { −76 −26 −4 −7 12 51 5 24 7 −17 −16 −12 [[ −5 4 2 13 ]] } { 5 723 5 69 −38 −8 −32 −15 −31 24 11 [[ 2 18 11 −15 ]] } },

-   -   Otherwise, if nTrS is equal to [[48]]        , lfnstTrSetIdx is equal to 2, and lfnstIdx is equal to 1 the        following applies:        -   lowFreqTransMatrix[m] [n]=lowFreqTransMatrixCol0to15[m] [n]            with m=0 . . . [[15]]            , n=0 . . . 15

lowFreqTransMatrixCol0to15= { { −121 33 4 4 1 2 [[ 0 1 ]] −1 −1 1 0 0 0[[ 0 0 ]] } { 0 −2 0 0 0 0 [[ 0 0 ]] 121 −23 −7 −3 −2 −1 [[ −1 0 ]] } {−20 19 −5 2 −1 1 [[ 0 0 ]] 16 3 −2 0 0 0 [[ 0 0 ]] } { 32 108 −43 10 −93 [[ −3 1 ]] 4 19 −7 1 −1 0 [[ 0 0 ]] } { −3 0 −1 0 0 0 [[ 0 0 ]] −29 11−2 1 0 0 [[ 0 0 ]] } { −4 −12 −3 1 −1 0 [[ 0 0 ]] 19 105 −31 7 −6 1 [[−2 0 ]] } { 7 1 2 0 0 0 [[ 0 0 ]] 4 3 −2 0 0 0 [[ 0 0 ]] } { −8 −31 14−4 3 −1 [[ 1 0 ]] 9 43 0 1 −1 0 [[ 0 0 ]] } { −15 −43 −100 23 −12 6 [[−4 2 ]] −6 −17 −48 10 −5 2 [[ −1 1 ]] } { −3 1 2 0 0 0 [[ 0 0 ]] −6 3 10 0 0 [[ 0 0 ]] } { −1 −6 −3 2 −1 0 [[ 0 0 ]] −6 −35 9 0 2 0 [[ 0 0 ]] }{ −5 −14 −48 2 −5 1 [[ −2 0 ]] 10 24 99 −17 10 −4 [[ 3 −1 ]] } { −2 0 20 0 0 [[ 0 0 ]] −2 0 1 0 0 0 [[ 0 0 ]] } { −2 −10 −4 0 0 0 [[ 0 0 ]] 311 −1 −1 0 0 [[ 0 0 ]] } { −2 −3 −25 −2 −3 0 [[ −1 0 ]] −1 −3 −1 4 −2 2[[ 0 1 ]] } { 4 −4 28 103 −42 24 [[ −9 7 ]] 1 2 4 0 3 −1 [[ 0 0 ]] } },

-   -   -   lowFreqTransMatrix[m]            [n]=lowFreqTransMatrixCol16to31[m−[[16]]            ] [n] with m=[[16]]            . . . [[31]]            , n=0 . . . 15

lowFreqTransMatrixCol16to31= { { 24 −5 −1 −1 0 0 [[ 0 0 ]] 5 −1 0 0 0 0[[ 0 0 ]] } { 17 1 −2 0 0 0 [[ 0 0 ]] −27 4 2 0 0 0 [[ 0 0 ]] } { −12014 8 1 3 1 [[ 1 0 ]] −18 −2 3 0 1 0 [[ 0 0 ]] } { 11 −30 9 −2 1 −1 [[ 00 ]] 0 −8 2 0 0 0 [[ 0 0 ]] } { 12 7 −1 0 0 0 [[ 0 0 ]] −117 12 9 1 3 0[[ 1 0 ]] } { 9 46 −6 0 0 0 [[ 0 0 ]] 8 −29 9 −3 1 0 [[ 0 0 ]] } { 22 −81 −1 0 0 [[ 0 0 ]] −28 −9 4 0 1 0 [[ 0 0 ]] } { −13 −105 17 −2 2 0 [[ 00 ]] −8 −25 −3 0 0 0 [[ 0 0 ]] } { 1 −5 19 −6 3 −1 [[ 1 0 ]] 2 7 15 −3 1−1 [[ 0 0 ]] } { 0 3 −2 0 0 0 [[ 0 0 ]] −20 8 −2 0 0 0 [[ 0 0 ]] } { 1−6 11 −2 2 0 [[ 1 0 ]] −9 −100 17 −1 1 0 [[ 0 0 ]] } { 4 14 32 0 2 0 [[1 0 ]] −4 0 −39 6 −4 1 [[ −1 0 ]] } { −1 −1 1 −1 0 0 [[ 0 0 ]] −1 −4 2 00 0 [[ 0 0 ]] } { −6 −40 −15 6 −2 1 [[ 0 0 ]] 5 57 −6 2 0 0 [[ 0 0 ]] }{ −7 −8 −97 17 −9 3 [[ −3 1 ]] −8 −26 −61 −1 −3 −1 [[ −1 −1 ]] } { −1 0−9 −42 17 −9 [[ 3 −2 ]] −1 1 −14 6 −4 2 [[ −1 0 ]] } },

-   -   -   lowFreqTransMatrix[m]            [n]=lowFreqTransMatrixCol32to47[m−[[32]]            ] [n] with m=[[32]]            . . . [[47]]            , n=0 . . . 15

lowFreqTransMatrixCol32to47= { { 3 −1 0 0 2 −1 0 0 2 −1 0 0 [[ 1 0 0 0]] } { −12 2 1 0 −5 1 0 0 −1 0 0 0 [[ −2 0 0 0 ]] } { 17 −3 −1 0 6 −1 −10 2 0 0 0 [[ 2 0 0 0 ]] } { −7 −1 2 0 −3 −1 1 0 −2 −2 1 0 [[ 0 0 0 0 ]]} { −32 −3 3 0 12 −2 −1 0 7 0 0 0 [[ 1 0 0 0 ]] } { −3 −19 3 0 −4 −6 1 00 0 0 0 [[ 0 −1 0 0 ]] } { 117 −10 −8 0 32 1 −4 0 3 1 −1 0 [[ −3 1 0 0]] } { −7 32 −5 1 −1 4 0 0 2 −1 0 0 [[ 1 0 −1 0 ]] } { 4 10 5 −1 0 3 1 0−2 1 2 0 [[ −1 1 1 0 ]] } { 30 13 −3 0 −116 6 10 0 −35 −5 4 0 [[ −3 −1 00 ]] } { −10 −63 1 2 −17 3 −4 0 −1 9 −1 0 [[ 3 4 −1 0 ]] } { 2 −3 −4 0 2−2 −2 0 0 0 −1 0 [[ 0 −1 −1 0 ]] } { −8 −2 −1 1 30 4 −4 1 −102 4 8 −1 [[−69 −2 6 −1 ]] } { 1 −95 18 −6 −10 −34 −2 0 −4 17 −2 0 [[ 0 2 1 0 ]] } {2 10 24 −7 5 9 19 −1 0 1 4 0 [[ −2 0 1 0 ]] } { −1 −2 −4 4 0 3 1 −1 0 20 −2 [[ 2 0 0 0 ]] } },

-   -   Otherwise, if nTrS is equal to [[48]]        , lfnstTrSetIdx is equal to 2, and lfnstIdx is equal to 2 the        following applies:        -   lowFreqTransMatrix[m] [n]=lowFreqTransMatrixCol0to15[m] [n]            with m=0 . . . [[15]]            , n=0 . . . 15

lowFreqTransMatrixCol0to15= { { 87 −41 3 −4 1 −1 [[ 0 −1 ]] −73 28 2 1 11 [[ 0 0 ]] } { −75 4 7 0 2 0 [[ 1 0 ]] −41 36 −7 3 −1 1 [[ 0 0 ]] } {26 −44 22 −6 4 −2 [[ 1 −1 ]] 77 24 −22 2 −4 0 [[ −1 0 ]] } { −39 −68 37−7 6 −2 [[ 2 0 ]] −9 56 −21 1 −2 0 [[ −1 0 ]] } { 10 −20 2 0 1 0 [[ 0 0]] 50 −1 8 −5 1 −1 [[ 0 0 ]] } { −21 −45 8 −2 3 −1 [[ 1 0 ]] −7 −30 26−8 3 −1 [[ 1 −1 ]] } { −4 −2 −55 28 −8 5 [[ −3 2 ]] −2 37 43 −19 1 −2 [[1 −1 ]] } { 2 19 47 −23 6 −4 [[ 2 −1 ]] −23 −22 −44 17 −2 2 [[ −1 0 ]] }{ −19 −62 −9 3 0 0 [[ 0 0 ]] −12 −56 27 −7 3 −1 [[ 1 0 ]] } { 1 9 −5 0−1 0 [[ 0 0 ]] 0 22 −1 2 0 1 [[ 0 0 ]] } { 5 17 −9 0 −2 1 [[ 0 0 ]] 1354 −2 7 −1 1 [[ 0 0 ]] } { 7 27 56 −2 10 −3 [[ 3 −1 ]] −2 −6 8 −28 3 −4[[ 1 −1 ]] } { 0 0 19 −4 3 −2 [[ 2 −1 ]] −3 −13 10 −4 1 0 [[ 0 0 ]] } {−3 0 −27 −80 40 −16 [[ 6 −4 ]] 4 3 31 61 −22 7 [[ −1 1 ]] } { 1 2 −8 6−1 1 [[ 0 0 ]] 2 8 −5 −1 0 0 [[ 0 0 ]] } { −4 −18 −57 8 −8 1 [[ −3 0 ]]−5 −20 −69 7 −6 2 [[ −2 1 ]] } },

-   -   -   lowFreqTransMatrix[m]            [n]=lowFreqTransMatrixCol16to31[m−[[16]]            ] [n] with m=[[16]]            . . . [[31]]            , n=0 . . . 15

lowFreqTransMatrixCol16to31= { { 30 −5 −6 1 −1 0 [[ 0 0 ]] −8 −3 3 0 0 0[[ 0 0 ]] } { 72 −29 −2 0 −1 0 [[ −1 0 ]] −37 6 7 −2 1 0 [[ 0 0 ]] } { 7−38 10 0 1 0 [[ 0 0 ]] −51 27 4 −3 2 −1 [[ 1 0 ]] } { −45 4 −3 6 −1 2 [[0 1 ]] 49 −13 3 −3 −1 0 [[ 0 0 ]] } { 66 17 −24 4 −3 1 [[ −1 0 ]] 13 −4915 1 0 0 [[ 0 0 ]] } { −9 69 −33 5 −2 0 [[ −1 0 ]] −44 −31 10 7 −2 2 [[0 1 ]] } { −47 −34 −27 5 4 −1 [[ 1 0 ]] −39 −2 27 4 −2 1 [[ 0 0 ]] } {−33 3 22 −2 −4 1 [[ −1 0 ]] −58 −17 6 −6 7 −1 [[ 1 0 ]] } { 7 −8 16 −6 4−2 [[ 1 −1 ]] −15 54 −23 2 −1 0 [[ 0 0 ]] } { −13 17 0 −2 0 −1 [[ 0 0 ]]−46 −10 −10 4 −1 1 [[ 0 0 ]] } { 4 51 −3 −6 −1 −1 [[ 0 0 ]] −20 6 −34 9−2 2 [[ −1 0 ]] } { −1 −4 −68 35 −5 5 [[ −2 1 ]] 0 35 43 −4 −6 1 [[ −1 0]] } { −6 −37 −18 −5 2 −2 [[ 1 −1 ]] 6 −6 −7 25 −6 4 [[ −1 1 ]] } { −4−7 −26 −6 −10 6 [[ −4 1 ]] 3 8 14 −18 15 −5 [[ 2 −1 ]] } { 1 24 3 5 −1 1[[ 0 0 ]] −3 12 6 −10 1 −1 [[ 0 0 ]] } { 1 4 0 33 −7 5 [[ −2 1 ]] 0 −953 −22 3 −1 [[ 0 0 ]] } },

-   -   -   lowFreqTransMatrix[m]            [n]=lowFreqTransMatrixCol32to47[m−[[32]]            ] [n] with m=[[32]]            . . . [[47]]            , n=0 . . . 15

lowFreqTransMatrixCol32to47= {{3 2 −1 0 −2 −1 0 0 1 1 0 0 [[−1 0 0 0]]}{12 3 −4 0 −3 −2 1 0 4 0 0 0 [[−1 0 0 0]]} {31 −5 −8 3 −14 0 5 −1 6 1 −30 [[−4 −1 1 0]]} {−19 2 0 0 5 1 1 0 −2 0 −1 0 [[1 0 0 0]]} {−53 34 6 −530 −7 −11 3 −11 −2 5 1 [[4 2 −1 −1]]} {49 7 2 −6 −23 −3 −2 2 9 4 0 0[[−2 −1 −1 0]]} {−11 32 −8 −7 27 −12 −6 6 −13 0 4 −3 [[3 −1 −2 1]]} {−2340 −2 5 43 −11 −8 −1 −18 −4 5 2 [[4 3 0 −1]]} {−42 −25 4 6 34 8 2 −2 −15−1 0 −1 [[3 2 0 1]]} {−80 −27 20 −4 −66 23 −2 −2 20 −3 −2 3 [[−14 2 3−1]]} {16 −52 28 1 59 15 −8 −5 −28 −7 2 2 [[10 3 0 −1]]} {−14 −38 −12−10 9 5 7 6 −9 7 −4 −3 [[4 −4 0 3]]} {16 10 55 −24 15 46 −52 1 35 −43 1012 [[−23 13 5 −8]]} {−2 −4 −1 13 0 2 −4 −3 3 −1 2 1 [[−2 0 −2 −1]]} {−9−1 −25 10 45 −11 18 2 86 1 −13 −4 [[−65 −6 7 2]]} {4 −27 −2 −9 5 36 −135 −7 −17 1 2 [[4 6 4 −1]]}},

-   -   Otherwise, if nTrS is equal to [[48]]        , lfnstTrSetIdx is equal to 3, and lfnstIdx is equal to 1 the        following applies:        -   lowFreqTransMatrix[m] [n]=lowFreqTransMatrixCol0to15[m] [n]            with m=0 . . . [[15]]            , n=0 . . . 15

lowFreqTransMatrixCol0to15= {{−115 37 9 2 2 1 [[1 0]] 10 −29 8 0 1 0 [[10]]} {15 51 −18 0 −3 0 [[−1 0]] −95 7 34 −3 5 −1 [[2 0]]} {29 −22 16 −63 −2 [[1 −1]] −4 −80 12 15 0 3 [[0 1]]} {−36 −98 25 5 4 1 [[2 1]] −59 11−17 1 1 1 [[0 0]]} {−6 18 3 −3 −1 0 [[0 0]] −50 −5 −38 12 0 2 [[0 1]]}{4 15 52 −13 5 −3 [[2 −1]] −17 −45 16 24 −2 4 [[−1 2]]} {−20 −7 −43 4 01 [[−1 1]] −7 35 0 12 −4 1 [[−1 0]]} {4 29 1 26 −5 4 [[−2 1]] −17 −7 −736 6 2 [[1 1]]} {12 13 10 2 −1 3 [[−1 1]] 17 −2 −46 12 7 0 [[2 0]]} {5 2090 −17 4 −3 [[2 −1]] 6 66 8 28 −7 3 [[−1 1]]} {−3 −4 −34 −12 2 −1 [[−10]] 5 25 11 43 −10 4 [[−2 1]]} {−1 −3 2 19 −2 4 [[−1 2]] 9 3 −35 22 11 1[[2 0]]} {10 −4 −6 12 5 1 [[1 0]] 11 −9 −12 −2 −7 0 [[−1 0]]} {4 6 14 53−4 4 [[0 2]] 0 −1 −20 −13 3 2 [[−1 1]]} {2 9 13 37 19 6 [[2 2]] −9 −3 −9−28 −20 −4 [[−3 −1]]} {3 −3 12 84 −12 8 [[−2 3]] 6 13 50 −1 45 1 [[70]]}},

-   -   -   lowFreqTransMatrix[m]            [n]=lowFreqTransMatrixCol16to31[m−[[16]]            ] [n] with m=[[16]]            . . . [[31]]            , n=0 . . . 15

lowFreqTransMatrixCol16to31= {{23 −8 −8 1 −1 0 [[0 0]] 3 3 −2 −1 0 0 [[00]]} {23 −47 1 6 0 1 [[0 1]] 8 5 −12 0 −1 0 [[0 0]]} {45 7 −59 7 −2 1[[−1 0]] −15 41 −3 −16 2 −3 [[0 −1]]} {6 −13 7 −3 0 0 [[0 0]] 14 −4 −143 −1 0 [[0 0]]} {3 67 −7 −40 3 −6 [[1 −3]] −12 −13 65 −3 −10 0 [[−1 0]]}{−87 −8 −14 7 8 1 [[2 0]] 23 −35 −6 −3 1 1 [[0 0]]} {−51 −2 −57 5 15 0[[4 0]] 7 39 5 −55 1 −7 [[1 −3]]} {−5 21 −3 5 −1 −3 [[ 0 −1]] −11 2 −52−3 27 −2 [[5 0]]} {16 −45 −9 −53 6 1 [[1 0]] 70 16 8 −4 −37 1 [[−7 0]]}{29 5 −19 12 9 −1 [[ 1 0]] −10 14 −1 −13 7 0 [[1 0]]} {23 20 −40 12 21−3 [[4 −1]] 25 −28 −10 5 8 6 [[0 2]]} {−7 −65 −19 −22 11 4 [[2 1]] −75−18 3 −1 −10 2 [[0 1]]} {33 −10 −4 18 18 −4 [[4 −1]] 28 −72 1 −49 15 2[[2 1]]} {−3 1 −5 35 −16 −6 [[−1 −2]] 46 29 13 21 37 −5 [[4 −1]]} {1 189 28 24 6 [[2 2]] −20 −5 −25 −33 −36 9 [[−2 2]]} {−2 18 −22 −37 −13 14[[0 3]] 1 −12 −3 2 −15 −8 [[1 −1]]}},

-   -   -   lowFreqTransMatrix[m]            [n]=lowFreqTransMatrixCol32to47[m−[[32]]            ] [n] with m=[[32]]            . . . [[47]]            , n=0 . . . 15

lowFreqTransMatrixCol32to47= {{4 0 0 −1 1 1 0 0 2 0 0 0 [[0 0 0 0]]} {3−3 1 −1 2 1 −2 0 1 −1 0 0 [[1 1 −1 0]]} {1 0 7 −2 −3 6 1 −2 0 0 1 0 [[−12 0 −1]]} {2 8 −3 −5 2 0 0 0 0 3 0 −1 [[ 1 0 0 0]]} {9 −20 −5 22 −2 0 0−1 2 −3 −2 3 [[−1 0 1 0]]} {2 5 −17 0 3 −1 −1 −5 0 1 −4 0 [[1 0 0 −2]]}{1 −10 41 2 4 −3 −2 3 −1 −2 7 1 [[1 −1 −1 0]]} {0 27 8 −58 2 −5 25 3 0 30 −5 [[ 0 −2 7 0]]} {−12 29 3 21 4 0 5 −1 −3 4 1 4 [[2 0 1 0]]} {0 −6 13−4 0 −4 1 5 0 −1 −1 1 [[0 −1 0 0]]} {−4 21 −64 −8 −5 19 10 −48 3 −1 10−3 [[0 4 3 −6]]} {2 −35 −27 4 1 8 −17 −19 3 0 3 −6 [[0 2 −1 −2]]} {56−23 22 −1 4 −1 −15 26 6 4 −10 0 [[0 2 −3 2]]} {−10 −53 −18 8 9 12 −41−25 −2 2 13 −16 [[4 1 −5 1]]} {−13 42 1 57 −22 −2 −25 −28 5 6 19 −12[[−5 −3 −2 4]]} {19 14 −4 −12 −4 5 17 8 2 −4 −4 4 [[−2 2 1 0]]}},

-   -   Otherwise, if nTrS is equal to [[48]]        , lfnstTrSetIdx is equal to 3, and lfnstIdx is equal to 2 the        following applies:        -   lowFreqTransMatrix[m] [n]=lowFreqTransMatrixCol0to15[m] [n]            with m=0 . . . [[            , n=0 . . . 15

lowFreqTransMatrixCol0to15= {{109 −26 −8 −3 −2 −1 [[−1 0]] −50 28 2 1 00 [[0 0]]} {−39 31 −5 2 −1 1 [[0 0]] −95 6 18 0 4 0 [[1 0]]} {29 −3 −2−2 0 0 [[0 0]] 0 −41 9 0 2 0 [[1 0]]} {18 96 −23 2 −5 1 [[−2 0]] −10 610 −2 1 −1 [[1 0]]} {−29 −60 16 −2 3 −1 [[1 0]] −52 9 −17 5 −2 1 [[−11]]} {−23 −5 −15 5 −2 1 [[−1 1]] 2 79 −13 −4 −2 −1 [[−1 0]]} {−7 −3 12−3 3 −1 [[1 0]] −31 −62 8 7 0 2 [[0 1]]} {1 −26 5 0 1 0 [[1 0]] 24 −3 43−6 4 −2 [[1 −1]]} {11 14 6 −3 1 −1 [[1 0]] 10 −7 −9 3 −2 1 [[−1 0]]}{−10 −11 −47 3 −4 1 [[−1 0]] 5 28 11 −2 −1 0 [[0 0]]} {−8 −24 −99 11 −103 [[−4 1]] −5 −36 19 −26 4 −5 [[1 −2]]} {−5 1 −1 0 1 0 [[0 0]] −10 −14−6 8 0 1 [[0 0]]} {1 12 −20 21 −4 5 [[−2 2]] −5 −2 −75 9 −1 2 [[−1 1]]}{2 −9 −18 8 −3 3 [[−1 1]] 3 −25 −62 −6 0 −2 [[0 −1]]} {4 9 39 18 0 2 [[01]] −6 −16 −22 −37 5 −5 [[1 −2]]} {−7 −2 15 −6 1 −1 [[1 −1]] −11 −3 22−14 0 −2 [[1 −1]]}},

-   -   -   lowFreqTransMatrix[m]            [n]=lowFreqTransMatrixCol16to31[m−[[16]]12] [n] with            m=[[16]]12 . . . [[31]]23, n=0 . . . 15

lowFreqTransMatrixCol16to31= {{−18 −8 6 0 1 0 [[1 0]] 6 −2 −3 0 0 0 [[00]]} {32 −49 5 1 1 0 [[0 0]] 27 −1 −14 2 −2 1 [[−1 0]]} {86 4 −33 2 −6 1[[−2 0]] −32 58 1 −7 0 −2 [[0 −]]} {−14 26 2 −4 1 −1 [[0 0]] −43 −9 35−2 4 −1 [[1 0]]} {13 56 −2 −9 0 −2 [[0 −1]] −34 −18 41 0 3 0 [[1 0]]}{−9 1 5 −1 1 0 [[0 0]] −4 49 2 −14 1 −3 [[0 −1]]} {−75 9 −45 5 −1 1 [[−10]] 14 35 0 −23 2 −5 [[1 −2]]} {−7 −64 9 14 0 3 [[0 1]] −12 −4 5 3 −1 1[[0 0]]} {22 21 1 −21 2 −4 [[1 −2]] 92 1 53 0 −9 1 [[−2 0]]} {−12 −2 −382 0 1 [[0 0]] 16 38 11 −16 −1 −3 [[0 −2]]} {0 25 41 5 −3 1 [[0 0]] 10 −5−7 12 2 1 [[0 0]]} {−17 −2 7 −5 3 −1 [[0 0]] −16 13 3 31 −1 6 [[0 2]]}{−1 −2 −16 −4 0 −1 [[0 0]] −7 7 −31 0 3 0 [[0 0]]} {−6 −61 14 −51 2 −6[[0 −2]] −19 0 40 −7 −17 0 [[−3 0]]} {−5 15 63 9 −16 0 [[−3 0]] 18 42−18 27 15 1 [[3 1]]} {−18 −7 30 −9 −4 0 [[−1 0]] −35 23 23 10 −17 1 [[−30]]}},

-   -   -   lowFreqTransMatrix[m]            [n]=lowFreqTransMatrixCol32to47[m−[[32]]            ] [n] with m=[[32]]            . . . [[47]]            , n=0 . . . 15

lowFreqTransMatrixCol32to47= {{−3 2 1 −1 0 0 0 0 −2 0 0 0[[0 0 0 0]]} {35 −3 −2 4 1 −1 −1 2 0 0 0[[2 0 0 0]]} {−14 −8 20 0 −2 −3 0 4 −1 −1 00[[−1 1 0 0]]} {14 −40 1 10 2 1 −10 1 2 −4 −1 −1[[0 0 −1 0]]} {19 −36−10 13 3 6 −14 −1 3 1 −1 −3[[1 1 −1 −1]]} {−31 −14 56 −1 13 −37 −4 20 −22 −10 0[[2 −4 0 −1]]} {1 −8 32 −1 7 −12 −4 10 0 2 −6 −1[[2 0 0 −2]]} {8−59 −3 26 14 6 −58 6 −5 17 −7 −18[[3 3 −1 −5]]} {−21 −11 1 40 −5 −4 −245 −4 5 −6 −5[[0 0 0 −3]]} {12 −9 −22 7 −8 60 4 −36 −6 −15 54 7[[3 −7 −814]]} {−1 1 9 −3 −3 −14 −3 12 2 4 −13 −2[[−1 3 2 −4]]} {−93 −15 −46 −323 −19 0 −47 8 4 8 3[[2 3 0 0]]} {4 11 −12 4 −12 14 −50 −1 −8 32 −4−54[[2 0 30 −15]]} {13 −4 11 9 17 0 24 5 1 −12 4 28[[0 0 −15 8]]} {12−34 9 −24 4 28 −2 4 −11 −4 30 2[[5 −13 −4 18]]} {−19 53 6 48 −65 12 −1211 −8 −16 10 −21[[−2 −12 6 2]]}},

This embodiment gives an example for 16×16 RST (a.k.a LFNST) and 16×48RST matrices. The newly added parts on top of JVET-P2001 are highlightedin bold and Italic and Deleted texts are marked with double brackets(e.g., [[a]] denotes the deletion of the character “a”).Low Frequency Non-Separable Transformation Matrix Derivation ProcessInputs to this process are:

-   -   a variable nTrS specifying the transform output length,    -   a variable predModeIntra specifying the intra prediction mode        for LFNST set selection.        Output of this process is the transformation matrix        lowFreqTransMatrix.        The variable lfnstTrSetIdx is specified in Table 39 depending on        predModeIntra.

TABLE 39 Specification of lfnstTrSetIdx predModeIntra lfnstTrSetIdxpredModeIntra < 0 1 0 <= predModeIntra <= 1 0  2 <= predModeIntra <= 121 13 <= predModeIntra <= 23 2 24 <= predModeIntra <= 44 3 45 <=predModeIntra <= 55 2 56 <= predModeIntra <= 80 1The transformation matrix lowFreqTransMatrix is derived based on nTrS,linstTrSetIdx, and lfnst_idx as follows:

-   -   If nTrS is equal to 16, lfnstTrSetIdx is equal to 0, and        lfnst_idx is equal to 1, the following applies:

lowFreqTransMatrix[m][n]= [[{{108 −44 −15 1 −44 19 7 −1 −11 6 2 −1 0 −1−1 0} {−40 −97 56 12 −11 29 −12 −3 18 18 −15 −3 −1 −3 2 1} {25 −31 −1 7100 −16 −29 1 −54 21 14 −4 −7 2 4 0} {−32 −39 −92 51 −6 −16 36 −8 3 2218 −15 4 1 −5 2} {8 −9 33 −8 −16 −102 36 23 −4 38 −27 −5 5 16 −8 −6}{−25 5 16 −3 −38 14 11 −3 −97 7 26 1 55 −10 −19 3} {8 9 16 1 37 36 94−38 −7 3 −47 11 −6 −13 −17 10} {2 34 −5 1 −7 24 −25 −3 8 99 −28 −29 6−43 21 11} {−16 −27 −39 −109 6 10 16 24 3 19 10 24 −4 −7 −2 −3} {−9 −10−34 4 −9 −5 −29 5 −33 −26 −96 33 14 4 39 −14} {−13 1 4 −9 −30 −17 −3 −64−35 11 17 19 −86 6 36 14} {8 −7 −5 −15 7 −30 −28 −87 31 4 4 33 61 −5 −1722} {−2 13 −6 −4 −2 28 −13 −14 −3 37 −15 −3 −2 107 −36 −24} {4 9 11 31 49 16 19 12 33 32 94 12 0 34 −45} {2 −2 8 −16 8 5 28 −17 6 −7 18 −45 4036 97 −8} {0 −2 0 −10 −1 −7 −3 −35 −1 −7 −2 −32 −6 −33 −16 −112}},]]{{97 −54 −7 1 −54 32 6 −2 −4 5 −1 −1 0 −2 −1 1} {−57 −65 65 0 −30 33 −11−1 44 3 −26 2 −4 −1 4 0} {−14 59 −19 −9 −83 1 41 −1 47 −36 −10 8 6 3 −70} {−8 −32 −8 19 −18 −84 54 9 1 59 −26 −16 6 5 −9 1} {27 31 86 −55 −10−29 0 20 −40 −6 −21 20 19 5 −8 −5} {−42 −16 −5 15 −35 23 19 −14 −79 −537 −1 57 −12 −27 6} {3 35 5 −17 16 55 37 −36 −2 67 −36 −4 1 −50 8 22}{−8 23 −12 −10 −45 −7 −77 39 6 57 20 −31 18 −23 21 −2} {−16 −31 −30 −992 −6 15 −4 13 15 50 21 −22 −2 −15 8} {−19 −18 −51 −24 −14 −5 −25 −5 −40−15 −77 43 −1 −3 42 −18} {−12 6 19 16 −30 −8 13 2 −45 −10 23 −14 −98 −2933 8} {−5 −7 −13 −6 20 36 40 107 −6 −9 −14 −15 −1 −17 −7 −22} {9 −18 1 310 −36 −9 1 20 −43 −1 8 22 −103 −7 31} {−4 −5 −12 −33 −1 0 −15 −14 −14−29 −43 −95 −11 13 −39 32} {2 −9 8 −21 10 −5 31 −13 5 −21 21 −47 40 6 98−8} {2 0 3 −7 0 −4 −1 −32 3 −2 1 −26 −6 −33 −24 −113}},

-   -   Otherwise, if nTrS is equal to 16, lfnstTrSetIdx is equal to 0,        and lfnst_idx is equal to 2, the following applies:

lowFreqTransMatrix[m][n]= [[{{119 −30 −22 −3 −23 −2 3 2 −16 3 6 0 −3 2 10} {−27 −101 31 17 −47 2 22 3 19 30 −7 −9 5 3 −5 −1} {0 58 22 −15 −102 238 2 10 −13 −5 4 14 −1 −9 0} {23 4 66 −11 22 89 −2 −26 13 −8 −38 −1 −9−20 −2 8} {−19 −5 −89 2 −26 76 −11 −17 20 13 18 −4 1 −15 3 5} {−10 −1 −16 23 25 87 −7 −74 4 39 −5 0 −1 −20 −1} {−17 −28 12 −8 −32 14 −53 −6 −68−67 17 29 2 6 25 4} {1 −24 −23 1 17 −7 52 9 50 −92 −15 27 −15 −10 −6 3}{−6 −17 −2 −111 7 −17 8 −42 9 18 16 25 −4 2 −1 11} {9 5 35 0 6 21 −9 3444 −3 102 11 −7 13 11 −20} {4 −5 −5 −10 15 19 −2 6 6 −12 −13 6 95 69 −29−24} {−6 −4 −9 −39 1 22 0 102 −19 19 −32 30 −16 −14 −8 −23} {4 −4 7 8 4−13 −18 5 0 0 21 22 58 −88 −54 28} {−4 −7 0 −24 −7 0 −25 3 −3 −30 8 −76−34 4 −80 −26} {0 6 0 30 −6 1 −13 −23 1 20 −2 80 −44 37 −68 1} {0 0 −1 5−1 −7 1 −34 −2 3 −6 19 5 −38 11 −115}},]] {{110 −46 −14 −1 −41 5 9 2 −118 4 −1 −2 2 −1 0} {−49 −73 41 11 −61 11 26 0 35 26 −16 −9 8 1 −8 −1} {1−79 6 15 91 −1 −29 0 −7 19 −3 −6 −13 2 8 0} {−18 −16 −36 19 −16 −102 530 −20 11 33 −1 14 22 0 −11} {16 7 98 −12 13 −33 32 2 −55 −24 −10 9 6 8−8 −1} {32 23 28 −2 11 −50 −28 15 86 8 −53 −4 −4 6 1 −2} {12 27 −3 5 352 81 −3 18 71 19 −31 −4 −9 −29 −6} {0 −20 −23 25 19 −3 67 19 32 −84 −718 −26 −18 −6 3} {−9 −21 −16 −104 3 −30 15 −41 12 3 7 27 −24 −15 3 15}{−6 9 7 22 −22 −16 −10 9 −25 20 −12 −17 −101 −50 23 10} {−7 −4 −40 6 0−15 14 −31 −42 3 −99 −21 22 −6 −27 11} {−5 −3 −7 −30 3 18 −5 94 −15 24−22 49 8 −33 −35 −11} {6 −3 11 24 −1 −21 −25 −30 9 −7 30 5 38 −90 −4729} {3 7 −1 36 3 −3 23 −19 0 35 −14 84 19 −5 62 38} {0 −4 −1 −27 5 0 1534 −2 −9 −4 −59 46 −45 80 12} {−1 1 1 −5 1 4 −3 29 1 −2 8 −21 −8 35 −21115}},

-   -   Otherwise, if nTrS is equal to 16, lfnstTrSetIdx is equal to 1,        and lfnst_idx is equal to 1, the following applies:

lowFreqTransMatrix[m][n]= [[{{−111 39 4 3 44 11 −12 −1 7 −16 −5 2 3 −1 42} {−47 −27 15 −1 −92 43 20 −2 20 39 −16 −5 10 −5 −13 2} {−35 −23 4 4−17 −72 32 6 −59 18 50 −6 0 40 0 −13} {13 93 −27 −4 −48 13 −34 4 −52 111 10 3 16 −3 1} {−11 −27 1 2 −47 −4 −36 10 −2 −85 14 29 −20 −2 57 4} {0−35 32 −2 26 60 −3 −17 −82 1 −30 0 −37 21 3 12} {−17 −46 −92 14 7 −10−39 29 −17 27 −28 17 1 −15 −13 17} {4 −10 −23 4 16 58 −17 26 30 21 67 2−13 59 13 −40} {5 −20 32 −5 8 −3 −46 −7 −4 2 −15 24 100 44 0 5} {−4 −138 −18 −7 −42 −63 −6 33 34 −23 15 −65 33 −20 2} {−2 −10 35 −19 5 8 −4414 −25 25 58 17 7 −84 −16 −18} {5 13 18 34 11 −4 18 18 5 58 −3 42 −2 −1085 38} {−5 −7 −34 −83 2 −1 −4 −73 4 20 15 −12 4 −3 44 12} {0 4 −2 −60 59 42 34 5 −14 9 80 −5 13 −38 37} {−1 2 7 −57 3 −7 9 68 −9 6 −49 −20 6 −436 −64} {−1 0 −12 23 1 −4 17 −53 −3 4 −21 72 −4 −8 −3 −83}},]] {{100 −47−1 −4 −57 −2 16 1 1 21 3 −3 −4 −2 −7 −2} {−52 −3 13 −1 −77 64 8 −3 28 32−32 −3 7 −12 −15 8} {45 23 −18 −2 29 57 −45 −2 37 −41 −48 13 −6 −38 1018} {−24 −87 32 4 14 −7 20 −1 52 −48 6 0 −4 −16 34 −2} {10 −6 13 −6 64 237 −15 43 58 −18 −33 12 −18 −56 −9} {−3 37 −30 6 −33 −58 −11 22 74 −2 3112 27 −39 −7 −11} {−22 −37 −37 7 −4 −51 −8 5 −38 6 −60 9 −10 −50 −25 44}{−10 −49 −58 23 22 22 −49 38 17 44 15 25 11 45 −2 −6} {−11 −12 −12 −8−11 7 −30 6 −3 −32 20 −31 −73 −12 −64 −54} {−2 12 −43 22 11 29 62 30 − 721 29 −4 −55 −43 44 7} {5 7 67 16 8 −1 −16 33 7 17 30 60 −40 −8 −33 51}{−1 13 −16 −24 −9 −28 −1 −22 4 6 8 −20 −30 − 62 61 18 52} {−5 −8 17 −304 −13 −49 −54 −3 58 7 19 −27 −39 50 −33} {−7 −16 −26 − 62 3 25 −10 −24−11 −9 68 −19 26 −22 −19 60} {3 1 34 33 −2 −4 −47 43 −9 16 8 −90 10 −1926 21} {−2 1 12 −92 5 −5 7 80 −3 6 −26 4 1 1 14 −19}},

-   -   Otherwise, if nTrS is equal to 16, lfnstTrSetIdx is equal to 1,        and lfnst_idx is equal to 2, the following applies:

lowFreqTransMatrix[m][n]= [[{{88 −55 6 −3 −66 27 9 −2 11 11 −13 1 −2 −71 2} {−58 −20 27 −2 −27 75 −29 0 47 −42 −11 11 −9 −3 19 −4} {−51 23 −225 −63 3 37 −5 1 64 −35 −4 29 −31 −11 13} {−27 −76 49 −2 40 14 9 −17 −5636 −25 6 14 3 −6 8} {19 −4 −36 22 52 7 36 −23 28 −17 −64 15 −5 −44 48 9}{29 50 13 −10 1 34 −59 1 −51 4 −16 30 52 −33 24 −5} {−12 −21 −74 43 −1339 18 −5 −58 −35 27 −5 19 26 6 −5} {19 38 −10 −5 28 66 0 −5 −4 19 −30−26 −40 28 −60 37} {−6 27 18 −5 −37 −18 12 −25 −44 −10 −38 37 −66 45 40−7} {−13 −28 −45 −39 0 −5 −39 69 −23 16 −12 −18 −50 −31 24 13} {−1 8 24−51 −15 −9 44 10 −28 −70 −12 −39 24 −18 −4 51} {−8 −22 −17 33 −18 −45−57 −27 0 −31 −30 29 −2 −13 −53 49} {1 12 32 51 −8 8 −2 −31 −22 4 46 −39−49 −67 14 17} {4 5 24 60 −5 −14 −23 38 9 8 −34 −59 24 47 42 28} {−1 −5−20 −34 4 4 −15 −46 18 31 42 10 10 27 49 78} {−3 −7 −22 −34 −5 −11 −36−69 −1 −3 −25 −73 5 4 4 −49}},]] {{80 −54 4 −3 −71 34 10 −1 17 11 −15 1−3 −8 2 3} {45 12 −23 0 9 −74 40 −1 −41 62 4 −15 8 −5 −26 9} {51 −25 18−8 32 −25 −29 11 −8 −49 67 −8 −23 47 −11 −19} {−5 −69 30 5 57 25 8 −25−62 12 −36 1 6 13 12 12 7} {43 13 −39 15 36 −24 12 −12 15 −43 −28 27 3−35 74 −14} {−22 −35 −23 7 16 −8 73 −8 46 −23 −15 −42 −46 30 −10 24} {4334 −13 −17 48 38 −35 11 6 −6 −39 −24 −21 −16 −46 53} {−3 −32 63 −17 31−37 −14 0 64 27 3 −1 −5 −61 −8 −7} {20 35 12 21 18 22 13 −56 29 28 −1540 −25 34 −33 −61} {10 −4 −24 7 3 6 54 30 14 15 13 42 −4 6 62 −22 −6−40} {−17 −43 −56 44 −6 −16 −26 −22 −2 −28 8 28 7 −40 −66 −5} {−14 −19−36 −15 5 11 −24 43 −22 33 −11 −18 −74 −20 16 −59} {1 22 36 −25 −22 −426 −36 −44 −56 −17 −45 −11 −44 −24 −38} {−3 −15 −14 −44 −2 −31 −17 30 21−17 −66 −7 53 42 −23 −41} {−1 −6 −11 28 −8 −16 −59 −64 10 21 −9 −74 8 1831 0} {9 11 41 97 −3 −8 2 57 −5 −10 −30 −21 1 4 −7 −12}},

-   -   Otherwise, if nTrS is equal to 16, lfnstTrSetIdx is equal to 2,        and lfnst_idx is equal to 1, the following applies:

lowFreqTransMatrix[m][n]= [[{{−112 47 −2 2 −34 13 2 0 15 −7 1 0 8 −3 −10} {29 −7 1 −1 −108 40 2 0 −45 13 4 −1 8 −5 1 0} {−36 −87 69 −10 −17 −3326 −2 7 14 −11 2 6 8 −7 0} {28 −5 2 −2 −29 13 −2 0 103 −36 −4 1 48 −16−4 1} {−12 −24 15 −3 26 80 −61 9 15 54 −36 2 0 −4 6 −2} {18 53 69 −74 1424 28 −30 −6 −7 −11 12 −5 −7 −6 8} {5 −1 2 0 −26 6 0 1 45 −9 −1 0 −11328 8 −1} {−13 −32 18 −2 15 34 −27 7 −25 −80 47 −1 −16 −50 28 2} {−4 −13−10 19 18 46 60 −48 16 33 60 −48 1 0 5 −2} {15 33 63 89 8 15 25 40 −4 −8−15 −8 −2 −6 −9 −7} {−8 −24 −27 15 12 41 26 −29 −17 −50 −39 27 0 35 −6726} {−2 −6 −24 13 −1 −8 37 −22 3 18 −51 22 −23 −95 17 17} {−3 −7 −16 −2110 24 46 75 8 20 38 72 1 2 1 7} {2 6 10 −3 −5 −16 −31 12 7 24 41 −16 −16−41 −89 49} {4 8 21 40 −4 −11 −28 −57 5 14 31 70 7 18 32 52} {0 1 4 11−2 −4 −13 −34 3 7 20 47 −6 −19 −42 −101}},]] {{96 −58 8 −5 52 −30 2 −1 04 −5 1 −6 6 −2 0} {52 2 −15 0 −63 54 −16 4 −68 35 2 −1 −17 0 7 −2} {−24−66 57 −10 −62 −38 47 −5 −20 −2 4 3 8 4 −7 3} {44 −1 −9 −1 −54 25 −2 169 −40 3 −2 62 −26 −5 1} {−19 −31 15 −1 3 29 −29 7 38 77 −58 5 28 32 −20−3} {−8 6 −4 1 32 −10 −3 −1 −63 9 7 0 102 −23 −11 0} {−15 −37 −59 50 −26−48 −50 48 −10 −21 −13 10 0 11 −1 −6} {26 54 −24 −1 −22 −44 38 −11 1 11−8 0 14 78 −43 1} {−8 −21 −26 16 0 −2 12 −11 22 43 73 −53 16 26 48 −30}{−12 −30 17 −6 18 55 −25 −1 −15 −62 27 −2 9 75 −22 −6} {11 16 36 70 1426 45 73 7 13 24 29 2 2 −5 4} {−14 −35 −51 20 9 25 32 −27 4 12 24 −10−16 −29 −77 40} {−4 −5 −16 −48 −4 −8 −16 11 12 23 53 83 6 12 18 50} {−9−23 −49 7 11 29 65 −21 −7 −17 −48 38 10 16 57 −8} {6 8 27 66 −5 −7 −24−58 −1 −3 −6 −4 7 15 32 76} {3 3 12 39 −4 −5 −18 −61 4 7 20 64 −5 −11−21 −74}},

-   -   Otherwise, if nTrS is equal to 16, lfnstTrSetIdx is equal to 2,        and lfnst_idx is equal to 2, the following applies:

lowFreqTransMatrix[m][n]= [[{{−99 39 −1 2 65 −20 −5 0 −15 −2 5 −1 0 3 −10} {58 42 −33 3 33 −63 23 −1 −55 32 3 −5 21 −2 −8 3} {−15 71 −44 5 −58−29 25 3 62 −7 −4 −4 −19 4 0 1} {46 5 4 −6 71 −12 −15 5 52 −38 13 −2 −6323 3 −3} {−14 −54 −29 29 25 −9 61 −29 27 44 −48 5 −27 −21 12 7} {−3 3 69−42 −11 −50 −26 26 24 63 −19 −5 −18 −22 12 0} {17 16 −2 1 38 18 −12 0 621 −14 5 89 −42 8 −2} {15 54 −8 6 6 60 −26 −8 −30 17 −38 22 −43 −45 42−7} {−6 −17 −55 −28 9 30 −8 58 4 34 41 −52 −16 −36 −20 16} {−2 −1 −9 −797 11 48 44 −13 −34 −55 6 12 23 20 −11} {7 29 14 −6 12 53 10 −11 14 59−15 −3 5 71 −54 13} {−5 −24 −53 15 −3 −15 −61 26 6 30 −16 23 13 56 44−35} {4 8 21 52 −1 −1 −5 29 −7 −17 −44 −84 8 20 31 39} {−2 −11 −25 −4 −4−21 −53 2 −5 −26 −64 19 −8 −19 −73 39} {−3 −5 −23 −57 −2 −4 −24 −75 1 39 −25 6 15 41 61} {1 1 7 18 1 2 16 47 2 5 24 67 3 9 25 88}},]] {{−71 51−8 3 72 −47 3 −1 −32 11 8 −2 9 2 −6 1} {−62 −12 30 −3 16 51 −41 5 40 −6118 3 −26 22 6 −7} {−17 61 −38 6 −45 −30 32 0 73 −11 −10 −5 −34 4 5 2}{−71 −18 16 4 −38 28 12 −8 5 52 −46 10 26 −49 18 3} {19 −40 13 −1 64 −188 −7 40 32 −39 4 −68 −16 26 3} {6 22 57 −39 −24 −37 −59 44 17 43 27 −20−11 −15 −10 2} {−29 −18 7 1 −44 −10 18 −3 −64 15 6 −2 −80 45 −2 1} {−21−56 12 3 1 −32 30 −7 43 21 10 −19 35 56 −59 13} {8 13 41 −12 −10 −33 −8−9 −1 −13 −54 47 30 60 43 −38} {14 46 −16 −6 20 71 −15 5 7 60 −26 −5 −455 −26 2} {−7 −10 −14 −65 8 8 38 74 −12 −28 −39 −42 10 7 16 14} {12 3261 −12 1 6 24 −28 −12 −35 −42 10 −13 −24 −63 44} {0 −5 −9 57 −10 −20 −56−2 −9 −14 −48 −58 7 16 17 54} {10 23 58 17 7 20 52 −24 4 10 38 −57 11 1357 8} {5 5 25 71 3 3 18 59 −4 −6 −18 −17 −7 −13 −32 −70} {−2 −2 −12 −40−4 −5 −19 −63 −5 −8 −23 −67 −4 −11 −19 −69}},

-   -   Otherwise, if nTrS is equal to 16, lfnstTrSetIdx is equal to 3,        and lfnst_idx is equal to 1, the following applies:

lowFreqTransMatrix[m][n]= [[{{−114 37 3 2 −22 −23 14 0 21 −17 −5 2 5 2−4 −1} {−19 −41 19 −2 85 −60 −11 7 17 31 −34 2 −11 19 2 −8} {36 −25 18−2 −42 −53 35 5 46 −60 −25 19 8 21 −33 −1} {−27 −80 44 −3 −58 1 −29 19−41 18 −12 −7 12 −17 7 −6} {−11 −21 37 −10 44 −4 47 −12 −37 −41 58 18 10−46 −16 31} {15 47 10 −6 −16 −44 42 10 −80 25 −40 21 −23 −2 3 −14} {1325 79 −39 −13 10 31 −4 49 45 12 −8 3 −1 43 7} {16 11 −26 13 −13 −74 −20−1 5 −6 29 −47 26 −49 54 2} {−8 −34 −26 7 −26 −19 29 −37 1 22 46 −9 −8137 14 20} {−6 −30 −42 −12 −3 5 57 −52 −2 37 −12 6 74 10 6 −15} {5 9 −642 −15 −18 −9 26 15 58 14 43 23 −10 −37 75} {−5 −23 −23 36 3 22 36 40 27−4 −16 56 −25 −46 56 −24} {1 3 23 73 8 5 34 46 −12 2 35 −38 26 52 2 −31}{−3 −2 −21 −52 1 −10 −17 44 −19 −20 30 45 27 61 49 21} {−2 −7 −33 −56 −4−6 21 63 15 31 32 −22 −10 −26 −52 −38} {−5 −12 −18 −12 8 22 38 36 −5 −15−51 −63 −5 0 15 73}},]] {{−106 50 3 3 −19 −20 20 −2 24 −27 −1 4 4 −1 −71} {14 29 −14 1 −71 73 3 −8 −20 −24 50 −9 12 −30 8 11} {39 −25 8 2 −28−30 43 −2 33 −76 −1 26 5 10 −51 14} {20 39 −46 6 −49 10 −12 6 6 13 −70 5−16 53 −12 −37} {33 73 −41 −1 68 9 21 −23 33 −12 30 1 −12 13 3 15} {−30−20 −21 21 36 52 −44 0 −25 −14 2 28 −13 10 −79 11} {7 33 19 −2 3 −13 606 −83 20 −10 50 −28 −15 −13 5} {−4 −4 44 −4 −22 28 14 4 42 53 16 22 −3153 −3 60} {−17 −32 20 −17 20 50 51 −42 0 −13 6 −1 −1 41 15 −70} {3 8 9−23 −20 −39 −13 −44 −18 15 38 −63 −52 10 −50 −16} {−17 −29 −48 −4 2 −820 −9 −49 −27 −7 −37 4 54 32 62} {−11 −36 −58 53 −19 −27 7 −11 14 19 4644 −42 −7 25 −22} {7 17 35 30 9 −4 −21 75 −18 −43 34 −13 −34 43 23 −26}{5 13 1 26 −6 −27 0 −2 −18 33 44 10 90 47 −26 −15} {−4 −14 −23 33 8 1964 57 17 32 −5 −60 1 −22 −36 −5} {9 13 40 98 2 5 −1 −54 −5 −9 −31 −24 −2−6 9 15}},

-   -   Otherwise, if nTrS is equal to 16, lfnstTrSetIdx is equal to 3,        and lfnst_idx is equal to 2, the following applies:

lowFreqTransMatrix[m][n]= [[{{−102 22 7 2 66 −25 −6 −1 −15 14 1 −1 2 −21 0} {12 93 −27 −6 −27 −64 36 6 13 5 −23 0 −2 6 5 −3} {−59 −24 17 1 −62−2 −3 2 83 −12 −17 −2 −24 14 7 −2} {−33 23 −36 11 −21 50 35 −16 −23 −7816 19 22 15 −30 −5} {0 −38 −81 30 27 5 51 −32 24 36 −16 12 −24 −8 9 1}{28 38 8 −9 62 32 −13 2 51 −32 15 5 −66 28 0 −1} {11 −35 21 −17 30 −1831 18 −11 −36 −80 12 16 49 13 −32} {−13 23 22 −36 −12 64 39 25 −19 23−36 9 −30 −58 33 −7} {−9 −20 −55 −83 3 −2 1 62 8 2 27 −28 7 15 −11 5}{−6 24 −38 23 −8 40 −49 0 −7 9 −25 −44 23 39 70 −3} {12 17 17 0 32 27 212 67 11 −6 −10 89 −22 −12 16} {2 −9 8 45 7 −8 27 35 −9 −31 −17 −87 −23−22 −19 44} {−1 −9 28 −24 −1 −10 49 −30 −8 −7 40 1 4 33 65 67} {5 −12−24 −17 13 −34 −32 −16 14 −67 −7 9 7 −74 49 1} {2 −6 11 45 3 −10 33 55 8−5 59 4 7 −4 44 −66} {−1 1 −14 36 −1 2 −20 69 0 0 −15 72 3 4 5 65}},]]{{−98 14 18 1 67 −29 −16 1 −6 23 −1 −4 0 −6 5 1} {−16 56 −11 −6 −46 −6333 9 49 −2 −46 −2 −10 23 16 −9} {−66 −25 −2 12 −35 44 14 −15 29 −70 8 176 20 −32 −9} {8 59 −34 −1 31 0 47 −7 −69 −43 −3 23 24 0 −22 −14} {−13 6224 −2 −49 −3 −23 2 −4 2 67 −9 3 −53 −23 31} {0 15 −56 16 26 30 39 −37 4534 32 28 −52 −20 8 11} {31 −6 23 16 28 −55 −25 −17 26 −14 16 38 −22 15−79 −19} {−22 −52 −29 18 −44 −40 26 −17 −30 41 −8 23 27 −51 −22 −22} {1−9 −41 19 0 −34 −14 −37 4 2 42 −11 59 58 15 51} {5 −36 −2 −29 24 −27 5342 15 −31 2 −24 −7 −30 −22 72} {−11 7 −74 −21 −4 13 −38 34 −1 19 −10 −54−13 13 −59 −14} {−1 −4 7 −64 5 5 29 42 25 28 55 30 40 24 1 −45} {13 −5−21 −4 26 −22 −13 −25 37 −49 16 −47 27 −54 30 −54} {12 25 16 25 21 39 7−2 47 29 −45 0 75 −22 −37 16} {−1 1 −41 −24 2 −4 −61 29 12 −25 −22 78 11−29 25 25} {3 −1 −10 91 5 −5 6 81 5 −6 26 3 1 3 16 −16}},

-   -   Otherwise, if nTrS is equal to 48, lfnstTrSetIdx is equal to 0,        and lfnst_idx is equal to 1 the following applies:        -   lowFreqTransMatrix[m] [n]=lowFreqTransMatrixCol0to15[m] [n]            with m=0 . . . 15, n=0 . . . 15

lowFreqTransMatrixCol0to15 = [[{ {−117 28 18 2 4 1 2 1 32 −18 −2 0 −1 00 0} {−29 −91 47 1 9 0 3 0 −54 26 −8 3 0 1 0 0} {−10 62 −11 −8 −2 −2 −1−1 −95 3 32 0 4 0 2 0} {−15 15 −10 −2 1 0 1 0 10 112 −20 −17 −4 −4 −1−2} {32 39 92 −44 4 −10 1 −4 26 12 −15 13 −5 2 −2 0} {−10 1 50 −15 2 −31 −1 −28 −15 14 6 1 1 1 0} {1 −33 −11 −14 7 −2 2 0 29 −12 37 −7 −4 0 −10} {0 6 −6 21 −4 2 0 0 −20 −24 −104 30 5 5 1 2} {−13 −13 −37 −101 29 −118 −3 −12 −15 −20 2 −11 5 −2 1} {6 1 −14 −36 9 −3 2 0 10 9 −18 −1 −3 1 00} {−12 −2 −26 −12 −9 2 −1 1 −3 30 4 34 −4 0 −1 0} {0 −3 0 −4 −15 6 −3 1−7 −15 −28 −86 19 −5 4 −1} {−1 9 13 5 14 −2 2 −1 −8 3 −4 −62 4 1 1 0} {62 −3 2 10 −1 2 0 8 3 −1 −20 0 1 0 0} {6 9 −2 35 110 −22 11 −4 −2 0 −3 1−18 12 −3 2} {−1 7 −2 9 −11 5 −1 1 −7 2 −22 4 −13 0 −1 0} }, ]] { {−11628 20 1 4 1 2 0 32 −20 −2 1 −1 0 0 0} {−23 −104 49 5 10 1 3 1 −24 25 −204 −1 1 0 0} {16 −31 −3 9 −1 2 0 1 104 −11 −32 0 −4 0 −2 0} {−20 12 −17 22 0 1 0 8 107 −19 −22 −3 −5 −1 −2} {33 38 83 −49 3 −11 1 −4 29 23 −17 14−9 2 −2 0} {−10 1 53 −20 3 −4 1 −1 −25 −12 16 7 −1 1 0 0} {−2 31 3 12 −5−1 −2 −1 −39 −8 −94 31 9 4 3 2} {1 21 18 −7 −4 −1 −1 0 −9 22 46 −23 0 −40 −1} {−12 −17 −36 −95 37 −9 11 −1 −11 −20 −11 2 −15 8 −1 2} {5 3 −16−33 14 −3 3 0 11 15 −17 −3 −5 1 0 0} {16 5 34 19 2 −3 −1 −2 6 −30 −3 −185 3 1 1} {−1 3 −2 11 11 −7 2 −1 8 23 37 90 −26 2 −7 −1} {1 −7 −16 −6 −125 −1 2 5 −5 −8 54 −9 −2 −2 −1} {8 15 0 38 97 −25 6 −7 −2 2 −12 −7 −6 14−3 2} {6 −7 −6 −12 2 0 1 1 17 2 16 −3 12 −1 0 −1} {0 7 0 −5 −48 14 −3 3−5 4 −23 −14 −11 −3 2 0} },

-   -   -   lowFreqTransMatrix[m] [n]=lowFreqTransMatrixCol16to31[m−16]            [n] with m=16 . . . 31, n=0 . . . 15

lowFreqTransMatrixCol16to31 = [[{ {14 −1 −3 0 −1 0 0 0 2 0 0 0 0 0 0 0}{33 5 −9 −1 −2 0 −1 0 −3 3 0 0 0 0 0 0} {32 −30 −4 4 −1 1 0 0 6 2 −5 0 00 0 0} {−20 −26 31 1 0 0 0 0 2 −16 −1 6 0 1 0 0} {29 −16 −22 8 0 1 0 1−20 6 4 −3 1 0 0 0} {−99 −4 9 5 5 2 2 1 44 −10 −11 1 −2 0 −1 0} {6 −99 326 −1 5 0 2 14 30 −27 −2 1 −1 0 −1} {−7 −46 10 −14 7 0 1 0 9 21 7 −6 −2−1 0 −1} {−12 10 26 12 −6 0 −1 0 −32 −2 11 3 3 −1 1 0} {38 26 −13 −1 −5−1 −1 0 102 3 −14 −1 −5 −1 −2 0} {−30 3 −92 14 19 0 3 0 −11 34 21 −33 1−2 0 −1} {−5 −17 −41 42 −6 2 −1 1 −1 −40 37 13 −4 2 −1 1} {−12 23 16 −11−17 0 −1 0 −11 97 −3 −3 0 −6 0 −2} {−4 4 −16 0 −2 0 1 0 34 23 6 −7 −4 −2−1 0} {−5 −4 −22 8 −25 3 0 0 −3 −21 2 −3 9 −2 1 0} {0 28 0 76 4 −6 0 −2−13 5 −76 −4 33 −1 3 0} }, ]] { {17 −1 −5 0 −1 0 0 0 1 0 0 0 0 0 0 0}{23 15 −8 −3 −2 −1 −1 0 −5 2 2 −1 0 0 0 0} {−43 32 7 −6 2 −1 1 0 −6 −5 61 0 0 0 0} {−19 −25 39 0 −1 0 0 0 2 −21 −2 8 0 1 0 0} {28 −21 −22 11 0 20 1 −24 7 5 −4 2 0 1 0} {−94 −8 9 8 5 3 2 1 48 −12 −15 1 −2 0 −1 0} {−1232 8 −37 8 −3 1 −1 1 −5 30 −3 −5 0 −1 0} {2 98 −8 −9 −6 −5 −1 −2 −12 −4221 9 0 3 0 1} {−7 11 40 11 −10 0 −2 0 −28 −1 5 7 4 −2 1 0} {37 27 −19 0−6 −2 −1 −1 95 15 −13 −6 −6 −2 −3 0} {38 −3 87 −20 −22 1 −4 0 20 −20 −2738 −2 0 0 0} {3 22 31 −32 14 −4 1 −1 −1 31 −31 −19 4 −1 1 −1} {17 −15−18 28 17 −2 0 0 8 −96 −17 6 8 9 1 3} {−7 8 −24 17 −29 4 0 1 −1 −7 −25−5 15 −4 2 0} {5 −18 −11 −39 5 6 1 1 42 −2 33 −2 −26 1 −3 0} {−1 33 6 497 −11 0 −3 6 32 −50 −7 24 −3 1 0} },

-   -   -   lowFreqTransMatrix[m] [n]=lowFreqTransMatrixCol32to47[m−32]            [n] with m=32 . . . 47, n=0 . . . 15

lowFreqTransMatrixCol32to47 = [[{ {3 0 −1 0 1 0 0 0 1 0 0 0 1 0 0 0} {72 −2 0 −1 1 0 0 2 1 −1 0 0 0 0 0} {6 −3 0 0 2 0 −1 0 2 −1 0 0 1 0 0 0}{1 −4 0 0 0 −3 0 1 0 −1 0 0 0 −2 0 0} {1 −4 −3 2 −4 1 0 0 1 −1 −2 1 −2 00 0} {−5 4 −3 0 8 −1 −2 0 −2 1 −1 0 4 0 −1 0} {−6 6 6 −3 1 3 −3 0 −1 1 10 0 1 −1 0} {2 2 5 −2 0 3 4 −1 0 0 1 0 0 1 2 −1} {11 −5 −1 6 −4 2 1 0 3−1 1 2 −1 0 0 0} {−29 10 10 0 10 −4 −1 1 −7 1 2 1 2 −1 0 0} {−9 −4 18 32 0 0 −2 −1 −1 3 0 0 0 0 −1} {−10 13 −1 −4 4 −4 3 4 −2 2 −1 −1 1 −1 1 2}{−21 −5 23 0 2 −2 −1 6 −3 −3 1 0 0 0 0 2} {108 −5 −30 6 −27 10 7 −2 11−3 −1 1 −4 1 0 1} {−7 1 3 −5 3 0 −1 0 0 1 0 −1 1 0 0 0} {9 18 −3 −35 −4−1 6 1 1 2 0 −3 −1 0 2 0} }, ]] { {4 0 −1 0 1 0 0 0 1 0 0 0 0 0 0 0} {53 −2 0 −1 1 1 0 2 1 −1 0 0 0 0 0} {−8 3 1 0 −2 −1 1 0 −3 1 0 0 −1 0 0 0}{2 −3 −1 0 1 −5 0 1 1 −1 0 0 0 −2 0 0} {1 −6 −3 3 −5 0 0 0 0 −2 −2 1 −20 0 0} {−5 6 −3 0 9 −1 −3 0 −1 1 −1 0 3 0 −1 0} {6 −6 −1 4 1 0 6 −2 1 −20 0 1 0 2 0} {2 −6 −11 4 0 −5 −1 2 1 −2 −2 1 1 −2 −1 1} {12 −5 −2 7 −4 31 0 3 −1 0 3 −1 1 0 0} {−37 10 16 0 9 −7 −1 2 −10 0 3 1 1 −2 0 1} {−2 7−18 −6 −1 1 0 1 −3 2 −4 0 −1 1 0 0} {7 −15 5 2 −4 2 −4 −5 1 −3 0 0 −1 0−1 −2} {3 22 −19 −11 1 1 5 −6 −1 6 0 −2 0 1 1 −2} {29 11 −4 −16 −7 3 6 03 −1 1 −2 −2 0 1 0} {86 2 −32 20 −26 10 4 −5 5 −2 −1 3 −6 1 0 0} {58 17−7 −26 −19 1 12 2 5 −3 0 −1 −3 −1 2 0} },

-   -   Otherwise, if nTrS is equal to 48, lfnstTrSetIdx is equal to 0,        and lfnst_idx is equal to 2 the following applies:        -   lowFreqTransMatrix[m] [n]=lowFreqTransMatrixCol0to15[m] [n]            with m=0 . . . 15, n=0 . . . 15

lowFreqTransMatrixCol0to15 = [[{ {−108 48 9 1 1 1 0 0 44 −6 −9 −1 −1 0−1 0} {55 66 −37 −5 −6 −1 −2 0 67 −30 −20 4 −2 0 −1 0} {2 86 −21 −13 −4−2 −1 −1 −88 5 6 4 5 1 1 0} {−24 −21 −38 19 0 4 −1 2 −23 −89 31 20 2 3 11} {9 20 98 −26 −3 −5 0 −2 −9 −26 15 −16 2 0 1 0} {−21 −7 −37 10 2 2 −11 −10 69 −5 −7 −2 −2 0 −1} {−10 −25 4 −17 8 −2 2 −1 −27 −17 −71 25 8 2 11} {2 5 10 64 −9 4 −3 1 −4 8 62 3 −17 1 −2 0} {−11 −15 −28 −97 6 −1 4 −17 3 57 −15 10 −2 0 −1} {9 13 24 −6 7 −2 1 −1 16 39 20 47 −2 −2 −2 0} {−711 12 7 2 −1 0 −1 −14 −1 −24 11 2 0 0 0} {0 0 7 −6 23 −3 3 −1 5 1 18 9613 −9 −1 −1} {−2 −6 −1 −10 0 1 1 0 −7 −2 −28 20 −15 4 −3 1} {−1 6 −16 024 −3 1 −1 2 6 6 16 18 −7 1 −1} {−5 −6 −19 −104 18 −4 3 0 6 0 35 −41 20−2 2} {−1 −2 0 23 −9 0 −2 0 1 1 8 −1 29 1 1 0} }, ]] { {−102 54 7 1 1 10 0 51 −10 −12 −1 −1 0 −1 0} {−60 −58 44 4 7 1 3 0 −55 43 15 −9 1 −1 1−1} {0 84 −25 −15 −4 −2 −1 −1 −86 7 5 6 6 1 2 1} {−33 −24 −30 22 1 5 0 2−31 −79 42 20 1 3 1 1} {7 25 88 −34 −6 −6 0 −2 −16 −17 10 −19 5 1 2 0}{23 28 17 1 −9 0 −2 0 33 −14 58 −22 −8 −1 −1 0} {−14 8 −45 21 −1 3 −1 14 72 25 −20 −8 −4 −1 −2} {−3 −10 −19 −58 15 −1 5 0 8 −2 −60 5 22 −1 2 1}{9 13 22 −12 6 −3 1 −1 16 43 15 40 −10 −4 −4 −2} {−15 −18 −30 −93 12 4 72 3 1 48 −18 11 −3 −2 −1} {−6 11 17 15 9 −5 −1 −2 −18 −4 −33 27 1 −3 10} {2 −2 12 −8 24 −4 3 −2 10 2 31 83 5 −14 −5 −3} {3 14 6 9 1 −1 −2 0 913 36 −10 18 −6 0 −2} {−2 −8 13 −5 −48 8 −2 2 −4 −6 −7 −27 −20 14 2 2}{−7 −6 −8 −24 −95 19 3 4 1 9 0 47 −34 15 −2 −1} {2 2 0 −26 17 −4 4 −1 −1−4 −13 −5 −38 −1 1 2} },

-   -   -   lowFreqTransMatrix[m] [n]=lowFreqTransMatrixCol16to31[m−16]            [n] with m=16 . . . 31, n=0 . . . 15

lowFreqTransMatrixCol16to31 = [[{ {9 −9 −1 1 0 0 0 0 3 −1 1 0 0 0 0 0}{−31 −19 14 4 1 1 1 0 −6 3 5 −2 0 0 0 0} {14 −5 0 3 0 0 0 0 10 −5 −2 0−1 0 0 0} {−30 26 36 −8 −2 −2 0 −1 14 18 −7 −9 −1 −1 0 0} {−61 −3 −2 3 71 1 0 12 16 −6 −1 0 −1 0 0} {−93 2 19 0 3 0 2 0 17 4 0 0 −1 0 0 0} {−4−66 28 36 −5 3 0 1 −10 20 33 −13 −8 0 0 −1} {−3 −75 5 −14 1 4 0 1 −36 318 −4 4 0 1 0} {−1 −27 13 6 1 −1 0 0 −34 −6 0 3 4 1 2 0} {28 23 76 −5−25 −3 −3 −1 6 36 −7 −39 −4 −1 0 −1} {−20 48 11 −13 −5 −2 0 −1 −105 −1917 0 6 2 3 0} {−21 −7 −42 14 −24 −3 0 0 11 −47 −7 3 −5 9 1 2} {−2 −32 −2−66 3 7 1 2 −11 13 −70 5 43 −2 3 0} {−3 11 −63 9 4 −5 2 −1 −22 94 −4 −6−4 −4 1 −2} {−2 10 −18 16 21 3 −2 0 −2 11 6 −10 6 −3 −1 0} {3 −6 13 7630 −11 −1 −2 −26 −8 −69 7 −9 −7 3 −1} }, ]] { {7 −11 0 2 0 0 0 0 2 −1 20 0 0 0 0} {37 15 −22 −3 −1 −1 −1 0 5 −7 −4 4 0 0 0 0} {25 −6 −2 4 −1 −10 0 10 −9 −2 1 −1 0 0 0} {−21 37 33 −15 −3 −3 0 −1 17 17 −14 −9 1 −1 00} {−67 4 −1 4 9 1 1 0 21 18 −11 0 0 −2 0 0} {45 50 −40 −35 5 −1 0 −1 −6−23 −32 15 11 1 1 1} {−72 27 −1 −20 8 0 2 0 19 −11 −16 9 5 0 0 0} {7 74−4 18 −5 −8 −2 −2 35 −8 −20 4 −7 1 −1 0} {25 14 72 −23 −28 −1 −3 −1 −330 −23 −42 8 3 1 0} {−7 −25 12 4 3 −1 0 0 −47 −4 0 7 8 1 2 0} {−29 39 −2−23 −10 −1 2 0 −95 −25 6 4 14 5 3 1} {−18 −21 −27 16 −29 0 2 1 29 −54 −63 −7 12 1 2} {3 48 12 55 −10 −14 −3 −3 −11 0 63 −12 −42 1 −1 0} {3 −4 623 0 4 −4 −1 12 −82 7 0 4 3 −2 2} {−5 14 −31 13 15 2 −2 −1 −6 25 5 −9 6−3 −1 0} {0 10 −5 −65 −22 22 4 3 32 0 64 −3 18 7 −8 −1} },

-   -   -   lowFreqTransMatrix[m] [n]=lowFreqTransMatrixCol32to47[m−32]            [n] with m=32 . . . 47, n=0 . . . 15

lowFreqTransMatrixCol32to47= { { 1 −1 0 0 1 0 0 0 0 −1 0 0 0 0 0 0 } { 71 −1 0 1 −1 −1 0 3 0 −1 0 0 0 0 0 } { 6 −6 1 1 2 −2 0 0 1 −2 0 0 1 −1 00 } { 2 2 −3 0 4 2 −3 −1 0 1 0 0 2 1 −1 0 } { 3 −4 −10 4 5 0 −2 2 0 −1−2 1 2 0 −1 0 } { −8 −4 5 9 −2 −2 −2 1 −2 0 1 1 −2 −1 −1 1 } { 4 −8 3 45 −5 0 1 0 −1 2 0 2 −2 0 1 } { −6 −16 6 10 2 −2 5 0 −3 −2 2 1 0 −1 2 0 }{ 1 −11 −21 8 −3 −2 −2 0 0 −4 −3 2 −1 −1 −1 −1 } { 2 11 0 8 −1 1 −5 2 42 −1 3 0 0 −2 1 } { −4 9 13 7 5 3 0 −4 6 −1 −1 0 2 0 0 −1 } { −2 0 23 −51 2 −1 −11 −2 3 1 −1 1 1 −1 −3 } { −8 17 −7 −42 2 −4 −12 1 0 −1 −3 −2 0−3 −3 1 } { −23 −12 20 1 4 12 1 −5 −4 4 −4 −1 1 2 −1 0 } { 6 9 0 −6 −2−5 −2 −4 1 −1 1 0 −1 −2 0 0 } { 16 40 12 −22 −2 −5 −17 −2 −3 −1 −1 6 −1−2 −3 1 }  },

-   -   Otherwise, if nTrS is equal to 48, lfnstTrSetIdx is equal to 1,        and lfnst_idx is equal to 1 the following applies:        -   lowFreqTransMatrix[m] [n]=lowFreqTransMatrixCol0to15[m] [n]            with m=0 . . . 15, n=0 . . . 15

lowFreqTransMatrixCol0to15 = [[{ {110 −49 −3 −4 −1 −1 0 −1 −38 −1 10 0 20 1 0} {−43 −19 17 −1 3 0 1 0 −98 46 14 −1 2 0 1 0} {−19 17 −7 3 −2 1 −10 −32 −59 29 3 4 0 2 0} {−35 −103 39 1 7 0 2 0 38 −13 25 −6 1 −1 0 0} {95 −6 −1 −1 0 −1 0 42 4 21 −11 1 −3 1 −1} {−5 −5 −28 9 −3 2 −1 1 −20 −7822 16 1 3 0 1} {14 17 27 −12 1 −3 1 −1 8 19 −13 4 −2 1 −1 0} {7 35 17 −4−1 0 0 0 3 8 54 −17 1 −2 1 −1} {−13 −27 −101 24 −8 6 −3 2 11 43 6 28 −63 −1 1} {−11 −13 −3 −10 3 −1 1 0 −19 −19 −37 8 4 2 0 1} {−4 −10 −24 −113 −2 0 −1 −6 −37 −45 −17 8 −2 2 −1} {−2 1 13 −17 3 −5 1 −2 3 0 −55 22 61 1 0} {3 1 5 −15 1 −2 1 −1 7 4 −7 29 −1 2 −1 1} {−4 −8 −1 −50 6 −4 2 −2−1 5 −22 20 6 1 0 0} {5 −1 26 102 −13 12 −4 4 −4 −2 −40 −7 −23 3 −5 1}{−5 −6 −27 −22 −12 0 −3 0 −5 8 −20 −83 0 0 0 0} }, ]] { {106 −50 −4 −4−1 −1 0 0 −46 4 11 1 2 0 1 0} {47 13 −19 1 −4 0 −2 0 88 −53 −10 2 −2 1−1 0} {25 −8 0 −2 0 −1 0 0 33 54 −37 −2 −5 −1 −2 0} {34 99 −49 −4 −8 −1−3 −1 −28 9 −13 3 1 1 0 0} {6 −18 4 −1 1 0 0 0 56 9 13 −13 0 −3 0 −1}{−4 −4 −22 10 −3 3 −1 1 −15 −77 35 14 2 4 1 1} {−16 −16 −16 13 0 3 0 1−11 −11 10 −2 2 −1 1 0} {−8 −37 −12 12 3 2 1 1 −5 −18 −56 26 0 4 0 2} {516 71 −33 6 −7 2 −3 −22 −40 −25 −1 7 2 2 1} {−21 −27 −65 21 1 6 1 2 −1013 −24 21 1 2 0 0} {3 4 18 2 −6 1 −2 0 9 39 52 −5 −10 −2 −3 −1} {−4 −14−27 10 0 4 0 2 −4 −10 49 −32 2 −3 1 −1} {5 −1 −8 −2 1 0 0 0 9 0 4 14 −41 −2 0} {−6 −14 −10 −27 7 0 2 0 −7 −7 −38 −4 7 3 1 1} {5 5 28 15 −6 −1−2 −1 5 −1 10 64 −18 3 −5 0} {10 10 25 87 −25 7 −7 1 −2 −3 −17 −3 −8 0 00} },

-   -   -   lowFreqTransMatrix[m] [n]=lowFreqTransMatrixCol16to31[m−16]            [n] with m=16 . . . 31, n=0 . . . 15

lowFreqTransMatrixCol16to31 = [[{ {−9 13 1 −2 0 0 0 0 −4 2 −3 0 0 0 0 0}{26 26 −15 −3 −2 −1 −1 0 11 −7 −9 2 0 0 0 0} {−72 43 34 −9 3 −2 1 −1 1336 −18 −10 0 −2 0 −1} {−1 7 6 −7 1 −1 0 0 −13 14 2 −4 2 −1 0 0} {21 70−32 −21 0 −4 −1 −1 34 − 26 −57 11 4 2 0 1} {80 −6 25 −5 −4 −1 −1 0 6 −247 −9 0 0 0 0} {48 −1 48 −15 −4 −2 −1 −1 1 60 −28 −42 5 −6 1 −2} {10 14−11 −34 4 −4 1 −1 −80 −7 −6 2 15 0 3 0} {−3 14 21 −12 −7 −2 −1 −1 −23 10−4 −12 3 0 1 0} {−12 −30 3 −9 5 0 1 0 −56 −9 −47 8 21 1 4 1} {17 14 −5814 15 0 2 0 −10 34 −7 28 4 −1 1 0} {8 74 21 40 −14 0 −2 0 −36 −8 11 −13−23 1 −3 0} {8 3 12 −14 −9 −1 −1 0 4 29 −15 31 10 4 1 1} {−16 −15 18 −29−11 2 −2 1 40 −45 −19 −22 31 2 4 1} {−1 5 8 −23 7 2 1 1 10 −11 −13 −3 12−3 2 0} {9 7 24 −20 41 3 6 1 15 20 12 11 17 −9 1 −2} }, ]] { {−8 15 0 −20 0 0 0 −4 1 −4 0 0 0 0 0} {−37 −21 21 3 2 1 1 0 −13 12 10 −4 0 −1 0 0}{57 −54 −30 14 −2 3 −1 1 −19 −36 28 12 −1 2 0 1} {−2 5 −9 4 0 0 0 0 20−19 −15 7 −2 2 −1 1} {17 63 −40 −21 0 −3 0 −1 20 −31 −52 16 6 3 1 1} {76−4 10 −9 −6 −2 −2 0 −1 −32 6 −3 2 1 1 0} {−63 6 −40 19 5 3 1 1 2 −48 3342 −7 5 −1 1} {−1 −9 19 31 −6 2 −2 1 70 1 3 −5 −17 0 −3 0} {−10 −31 −185 8 2 2 1 −29 −4 −27 16 13 1 2 0} {−11 −5 23 −15 −1 −2 0 −1 −66 2 −30 518 2 4 0} {−18 −34 60 −25 −9 1 −1 0 22 −28 −11 −17 7 3 1 1} {−10 −68 −36−6 18 4 4 1 22 8 −11 16 12 −2 1 0} {15 −13 −8 −15 5 −1 1 0 12 51 −20 386 −1 0 −1} {−13 −24 13 −38 6 3 1 1 39 −35 −30 −7 33 1 3 0} {−14 −2 9 −14−24 2 −2 0 −16 −44 −2 −4 1 10 1 2} {0 −4 −19 −12 −2 5 0 1 24 0 2 −21 180 2 0} },

-   -   -   lowFreqTransMatrix[m] [n]=lowFreqTransMatrixCol32to47[m−32]            [n] with m=32 . . . 47, n=0 . . . 15

lowFreaTransMatrixCol32to47 = [[{ {−2 2 0 1 −1 1 0 0 −1 1 0 0 −1 0 0 0}{9 −3 −1 2 3 −3 0 0 4 −1 0 0 2 −1 0 0} {3 0 −12 3 6 1 −3 2 1 −1 −2 0 3 1−1 1} {−2 11 −6 −2 −2 4 −3 0 0 3 −2 0 −1 1 −1 0} {−4 −32 5 24 1 −6 12 4−3 −2 4 −2 0 −1 0 0} {−7 3 13 −4 −3 5 1 −5 −2 3 1 −2 −1 2 −1 −2} {11 −11−51 11 −2 −10 −2 13 2 −6 −4 4 −2 −3 2 2} {−16 46 1 3 2 7 −24 0 2 −2 −5 81 −1 −2 2} {2 9 −10 0 1 −5 −4 4 2 −2 2 2 0 −2 1 0} {−11 −30 10 59 −2 841 8 2 5 6 −7 −1 3 5 −2} {23 34 −31 4 10 −22 −30 22 4 −15 9 20 2 −5 9 4}{−36 6 16 −14 2 19 −4 −12 −1 0 −7 −3 0 2 −2 −1} {61 22 55 14 13 3 −9 −651 −11 −21 −7 0 0 −1 3} {−25 41 0 12 9 7 −42 12 −3 −14 2 28 5 1 6 2} {−923 4 9 14 9 −14 −4 0 −12 −7 6 3 0 6 3} {−26 −1 18 −1 −12 32 3 −18 −5 10−25 −5 −2 1 −8 10} }, ]] { {−2 2 0 1 −1 0 0 0 −1 1 0 0 0 0 0 0} {−10 6 0−3 −4 4 −1 0 −4 1 0 0 −1 1 0 0} {−5 3 16 −6 −7 1 3 −4 −2 1 1 −1 −3 0 1−1} {0 −20 8 8 2 −6 8 1 −1 −3 4 −1 1 −1 1 −1} {−8 −29 7 26 −1 −6 14 5 −4−1 6 −2 −2 −1 1 −1} {−11 7 24 −3 −4 11 1 −10 −3 5 −1 −4 −1 4 −2 −1} {−316 52 −14 6 12 1 −20 0 6 −1 −7 3 2 −2 −1} {10 −48 3 1 −2 −7 33 0 −2 4 10−12 −2 3 2 −6} {−4 −18 17 40 3 13 30 0 3 5 3 −9 0 4 2 −4} {−5 −4 8 43 48 24 5 6 3 6 −6 1 1 3 −4 } {−6 −23 29 6 −1 20 25 −26 −4 11 −14 −22 −2 4−11 −2} {12 −13 −32 11 −5 −24 11 37 3 5 24 6 0 1 5 −4} {76 9 34 6 8 −10−12 −51 −1 −16 −19 −6 −3 −3 0 7} {−11 27 4 6 −3 9 −51 0 −8 −13 −12 44 51 11 18} {41 −2 −7 −7 6 −45 −15 16 6 −14 43 17 3 3 15 −19} {−7 43 −11 3333 9 11 −2 −1 −9 −16 −17 3 −7 1 3} },

-   -   Otherwise, if nTrS is equal to 48, lfnstTrSetIdx is equal to 1,        and lfnst_idx is equal to 2 the following applies:        -   lowFreqTransMatrix[m] [n]=lowFreqTransMatrixCol0to15[m] [n]            with m=0 . . . 15, n=0 . . . 15

lowFreqTransMatrixCol0to15 = [[{ {80 −49 6 −4 1 −1 1 −1 −72 36 4 0 1 0 00} {−72 −6 17 0 3 0 1 0 −3 58 −21 2 −3 1 −1 0} {−50 19 −15 4 −1 1 −1 1−58 −2 30 −3 4 −1 2 0} {−33 −43 28 −7 4 −2 2 −1 −38 11 −8 4 1 1 0 0} {1066 −21 −3 −3 0 −1 0 −53 −41 −2 16 −1 4 −1 1} {18 14 13 −9 2 −2 1 −1 3432 −31 12 −5 2 −2 1} {21 66 −1 9 −4 2 −1 1 −21 41 −30 −10 0 −2 0 −1} {1−6 −24 17 −5 3 −2 1 24 10 39 −21 5 −4 2 −1} {9 33 −24 1 4 0 1 0 6 50 261 −10 0 −2 0} {−7 −9 −32 14 −3 3 −1 1 −23 −28 0 −5 −1 0 0 0} {6 30 69−18 5 −4 3 −1 −3 −11 −34 −16 9 −4 2 −1} {1 −8 24 −3 7 −2 2 −1 −6 −51 −6−4 −5 0 −1 0} {4 10 4 17 −9 4 −2 1 5 14 32 −15 9 −3 2 −1} {−3 −9 −23 10−10 3 −3 1 −5 −14 −16 −27 13 −5 2 −1} {2 11 22 2 9 −2 2 0 −6 −7 20 −32−3 −4 0 −1} {2 −3 8 14 −5 3 −1 1 −2 −11 5 −18 8 −3 2 −1} }, ]] { {76 −494 −3 1 −1 0 0 −74 38 5 0 1 0 0 0} {−55 −4 19 1 3 0 1 0 −6 56 −28 1 −4 0−2 0} {−60 28 −9 4 0 1 0 0 −44 10 22 −2 2 −1 1 0} {−35 −14 25 −5 3 −1 20 −46 17 −11 7 1 1 0 0} {26 69 −29 −8 −5 −2 −2 −1 −26 −34 −9 22 −1 5 −11} {−22 12 −17 8 −2 2 −1 1 −56 −32 37 −5 5 0 2 0} {24 39 −27 11 −7 1 −20 16 26 0 −19 2 −3 0 −1} {−14 −52 −3 7 3 2 1 1 34 −24 45 −5 −1 −2 0 −1}{13 31 −10 −15 5 −3 1 −1 16 50 −2 4 −10 −1 −2 0} {−7 7 −36 11 2 2 0 1−21 −1 22 2 −8 1 −1 0} {−7 −29 −5 −5 9 0 2 0 −8 −34 −15 33 −12 6 −3 2}{−5 9 25 −18 1 −3 1 −1 −14 −1 −38 7 9 0 1 0} {15 29 80 −27 2 −7 0 −3 −4−37 −14 −9 4 1 2 0} {7 20 21 −11 3 −3 1 −1 12 44 31 11 −16 0 −4 0} {2 1629 −7 5 −3 0 −1 −12 −10 29 −41 −1 −2 1 0} {7 9 41 −3 −9 1 −2 0 1 −2 56−15 2 −3 0 −1} },

-   -   -   lowFreqTransMatrix[m] [n]=lowFreqTransMatrixCol16to31[m−16]            [n] with m=16 . . . 31, n=0 . . . 15

lowFreqTransMatrixCol16to31 = [[{ {26 0 −12 2 −2 1 −1 0 −7 −9 6 1 0 0 00 } {55 −46 −1 6 −2 1 −1 0 −22 7 17 −7 2 −1 1 0} {6 57 −34 0 −2 0 −1 034 −48 −2 14 −4 3 −1 1} {−55 24 26 −5 2 −1 1 0 15 46 −40 −1 −1 0 −1 0}{36 −5 41 −20 3 −3 1 −1 −30 26 −32 −3 7 −2 2 −1} {40 4 −4 −9 −3 −2 −1 −127 −31 −43 19 −2 3 −1 1} {−35 −17 −3 26 −6 5 −2 2 56 3 18 −25 −1 −2 −1−1} {33 32 −30 4 −3 −1 −1 0 −4 13 −16 −10 0 −1 0 0} {−27 1 −28 −21 16 −53 −2 −23 36 −2 40 −17 4 −3 1} {−36 −59 −24 14 4 2 1 1 −23 −26 23 26 −3 50 2} {−16 35 −35 30 −9 3 −2 1 −57 −13 6 4 −5 5 −1 1} {38 −1 0 25 6 2 1 147 20 35 1 −27 1 −5 0} {7 13 19 15 −8 1 −1 0 3 25 30 −18 1 −2 0 −1} {−1−13 −30 11 −5 2 −1 0 −5 −8 −22 −16 10 0 1 0} {13 −5 −28 6 18 −4 3 −1 −2627 −14 6 −20 0 −2 0} {12 −23 −19 22 2 0 1 0 23 41 −7 35 −10 4 −1 1} },]] { {27 2 −14 2 −2 1 −1 0 −6 −12 8 1 0 0 0 0} {53 −61 5 8 −1 2 0 0 −2717 21 −12 3 −2 1 −1} {30 38 −39 1 −3 0 −1 0 28 −57 8 15 −3 3 −1 1} {−1728 23 −10 0 −2 1 −1 18 33 −57 3 0 0 −1 0} {55 −19 32 −21 0 −3 0 −1 −30 1−27 5 8 −1 2 0} {−19 13 13 0 2 −1 1 0 −14 44 14 −22 3 −4 1 −1} {15 6 −3118 −5 2 −2 1 24 −13 −6 −13 0 0 0 0} {52 27 −23 −23 3 −4 0 −1 −51 4 −2421 2 2 1 0} {−9 20 −25 −24 12 −4 3 −1 −9 27 −26 30 −11 1 −2 0} {−42 −41−19 7 14 1 3 1 −25 9 33 39 −18 4 −3 1} {17 −23 17 −8 8 0 1 0 35 −15 −113 −10 0 −2 0} {−45 −2 −21 6 −4 2 0 1 −59 −35 −14 22 17 4 3 1} {21 21−18 28 −2 0 −1 0 0 12 27 −2 −22 3 −4 1} {−3 18 33 −32 3 −3 0 −1 −2 9 3 1−2 −3 0 0} {−3 −25 −31 −6 23 −3 4 0 −26 5 −22 3 −5 −1 1 0} {−5 −59 −3−25 2 4 1 1 46 −3 −12 18 21 −3 2 −1} },

-   -   -   lowFreqTransMatrix[m] [n]=lowFreqTransMatrixCol32to47[m−32]            [n] with m=32 . . . 47, n=0 . . . 15

lowFreqTransMatrixCol32to47 = [[{ {3 5 −1 −2 −2 −2 −1 1 1 1 0 0 −1 −1 00} {9 5 −12 1 −3 −4 4 2 4 1 −2 −1 −1 −1 1 0} {−10 7 21 −10 6 1 −11 0 −1−1 4 2 3 0 −2 −1} {17 −38 1 17 −3 11 15 −11 3 −1 −10 1 0 1 3 2} {15 −8 117 −1 −2 4 −8 2 0 −1 3 0 0 0 −1} {7 −49 52 10 −11 22 7 −26 −1 −6 −9 6 −22 4 −2} {−15 −13 −27 9 9 −6 20 5 −3 2 −6 −9 3 −3 1 5} {24 −26 −37 33 5−32 55 −5 −7 22 −14 −22 1 −9 −3 13} {43 −13 4 −41 −19 −2 −24 17 11 −4 84 −3 −3 −3 −3} {10 −26 38 7 −12 11 42 −22 −5 20 −14 −15 −1 −2 1 6} {2810 4 7 0 −15 7 −10 −1 7 −2 2 1 −3 0 0} {37 −37 −9 −47 −28 5 0 18 8 6 0−8 −4 −3 −3 1} {11 24 22 −11 −3 37 −13 −58 −5 12 −63 26 9 −15 11 8} {0−29 −27 6 −27 −10 −30 9 −3 −10 −7 77 9 −13 45 −8} {−76 −26 −4 −7 12 51 524 7 −17 −16 −12 −5 4 2 13} {5 7 23 5 69 −38 −8 −32 −15 −31 24 11 2 1811 −15} }, ]] { {2 7 0 −3 −2 −3 −2 1 1 1 1 0 −1 −1 0 0} {9 5 −20 3 −3 −67 4 3 1 −2 −3 −1 −1 1 1} {−8 12 22 −15 4 1 −15 1 0 −2 5 3 2 −1 −2 −2}{21 −52 16 24 −2 13 20 −22 2 −1 −14 3 1 1 4 2} {6 −4 21 10 −4 5 −6 −14 1−3 0 9 −1 1 2 −4} {6 29 −58 7 14 −30 8 23 3 6 9 −12 2 −3 −5 3} {−1 −35−27 36 9 −25 63 −5 −10 16 −11 −29 −1 −5 −4 14} {22 −3 4 11 −3 −15 17 −8−2 11 −1 −9 −3 −1 −5 4} {27 −17 −20 −27 −13 −23 −29 44 13 −16 38 4 −4 1−3 −12} {37 −22 36 −28 −14 16 22 −14 4 18 −16 −19 −3 −1 −8 9} {−19 −26 2−7 2 1 26 46 10 −3 47 −51 −11 15 −23 −8} {−5 26 16 40 23 −25 20 −1 0 230 −11 1 9 −6 −11} {36 −12 −2 −28 −10 −6 6 −3 −1 10 −6 −17 −4 −1 −13 5}{−18 38 16 −8 29 17 5 −19 −3 −4 −16 −59 −9 7 −38 20} {−60 −21 −10 10 −552 1 35 10 −15 −13 0 0 0 12 13} {14 5 8 11 11 −37 0 −20 −6 6 22 17 4 2 5−27} },

-   -   Otherwise, if nTrS is equal to 48, lfnstTrSetIdx is equal to 2,        and lfnst_idx is equal to 1 the following applies:        -   lowFreqTransMatrix[m] [n]=lowFreqTransMatrixCol0to15[m] [n]            with m=0 . . . 15, n=0 . . . 15

lowFreqTransMatrixCol0to15 = [[{ {−121 33 4 4 1 2 0 1 −1 −1 1 0 0 0 0 0}{0 −2 0 0 0 0 0 0 121 −23 −7 −3 −2 −1 −1 0} {−2 0 19 −5 2 −1 1 0 0 16 3−2 0 0 0 0 0} {32 108 −43 10 −9 3 −3 1 4 19 −7 1 −1 0 0 0} {−3 0 −1 0 00 0 0 −29 11 −2 1 0 0 0 0} {−4 −12 −3 1 −1 0 0 0 19 105 −31 7 −6 1 −2 0}{7 1 2 0 0 0 0 0 4 3 −2 0 0 0 0 0} {−8 −31 14 −4 3 −1 1 0 9 43 0 1 −1 00 0} {−15 −43 −100 23 −12 6 −4 2 −6 −17 −48 10 −5 2 −1 1} {−3 1 2 0 0 00 0 −6 3 1 0 0 0 0 0} {−1 −6 −3 2 −1 0 0 0 −6 −35 9 0 2 0 0 0} {−5 −14−48 2 −5 1 −2 0 10 24 99 −17 10 −4 3 −1} {−2 0 2 0 0 0 0 0 −2 0 1 0 0 00 0} {−2 −10 −4 0 0 0 0 0 3 11 −1 −1 0 0 0 0} {−2 −3 −25 −2 −3 0 −1 0 −1−3 −1 4 −2 2 0 1} {4 −4 28 103 −42 24 −9 7 1 2 4 0 3 −1 0 0} }, ]] {{120 −37 −5 −5 −2 −2 −1 −1 4 0 −1 0 0 0 0 0} {−2 −2 1 0 0 0 0 0 118 −26−8 −3 −3 −1 −1 0} {18 −22 6 −2 2 −1 0 0 −22 −2 4 1 1 0 0 0} {32 101 −468 −11 1 −4 0 0 28 −10 1 −2 0 −1 0} {−8 −15 7 −1 2 0 1 0 −32 9 0 1 0 0 00} {−5 −13 −1 2 0 1 0 0 19 94 −36 5 −8 0 −3 −1} {−7 −2 −1 1 0 0 0 0 −5−7 4 0 1 0 0 0} {10 37 −17 4 −4 0 −2 0 −11 −45 6 0 2 1 1 0} {5 4 6 −2 0−1 0 0 6 −14 7 −2 1 0 0 0} {−1 −3 −11 4 −2 1 0 0 −10 −44 9 2 2 1 1 0}{−16 −44 −93 33 −10 10 −2 4 −6 −17 −47 14 −4 4 −1 2} {−5 −8 −18 4 −1 1 01 7 19 55 −18 5 −5 1 −2} {−3 −12 −31 6 −3 3 0 0 10 23 71 −20 5 −5 1 −2}{−2 −11 3 0 2 0 1 0 3 10 −9 2 −1 1 0 0} {−5 −10 −38 9 −3 3 −1 1 2 7 17 1−1 0 −1 0} {0 −1 −5 −1 1 0 0 0 −2 −13 5 −1 2 −1 1 0} },

-   -   -   lowFreqTransMatrix[m] [n]=lowFreqTransMatrixCol16to31[m−16]            [n] with m=16 . . . 31, n=0 . . . 15

lowFreqTransMatrixCol16to31 = [[{ {24 −5 −1 −1 0 0 0 0 5 −1 0 0 0 0 0 0}{17 1 −2 0 0 0 0 0 −27 4 2 0 0 0 0 0} {−120 14 8 1 3 1 1 0 −18 −2 3 0 10 0 0} {11 −30 9 −2 1 −1 0 0 0 −8 2 0 0 0 0 0} {12 7 −1 0 0 0 0 0 −11712 9 1 3 0 1 0} {9 46 −6 0 0 0 0 0 8 −29 9 −3 1 0 0 0} {22 −8 1 −1 0 0 00 −28 −9 4 0 1 0 0 0} {−13 −105 17 −2 2 0 0 0 −8 −25 −3 0 0 0 0 0} {1 −519 −6 3 −1 1 0 2 7 15 −3 1 −1 0 0} {0 3 −2 0 0 0 0 0 −20 8 −2 0 0 0 0 0}{1 −6 11 −2 2 0 1 0 −9 −100 17 −1 1 0 0 0} {4 14 32 0 2 0 1 0 −4 0 −39 6−4 1 −1 0} {−1 −1 1 −1 0 0 0 0 −1 −4 2 0 0 0 0 0} {−6 −40 −15 6 −2 1 0 05 57 −6 2 0 0 0 0} {−7 −8 −97 17 −9 3 −3 1 −8 −26 −61 −1 −3 −1 −1 −1}{−1 0 −9 −42 17 −9 3 −2 −1 1 −14 6 −4 2 −1 0} }, ]] { {−23 5 1 1 0 0 0 0−6 2 0 0 0 0 0 0} {25 0 −4 0 −1 0 0 0 −28 6 2 0 0 0 0 0} {117 −16 −10 −2−4 −1 −2 0 24 2 −5 0 −1 0 0 0} {15 −31 10 −3 2 −1 1 0 −15 −9 5 0 1 0 10} {13 14 −4 0 −1 0 −1 0 −111 16 11 2 4 1 2 0} {10 60 −12 −1 −2 −1 −1 −114 −26 10 −4 2 −1 0 0} {−25 5 1 1 0 0 0 0 34 12 −7 −1 −2 −1 −1 0} {13 92−22 −1 −4 −1 −2 −1 10 50 −5 −2 −2 −2 −1 −1} {−1 −1 0 0 0 0 0 0 24 −26 30 0 0 0 0} {2 10 8 −2 1 −1 0 0 −12 −82 21 1 3 2 1 1} {1 −3 23 −7 4 −2 1−1 4 11 15 −4 2 −2 0 −1} {2 12 35 −8 2 −2 0 −1 −1 1 −11 5 −2 1 −1 0} {812 46 −9 3 −3 1 −1 −2 8 −19 4 −2 1 −1 0} {−6 −45 1 3 0 2 0 1 5 55 −1 −3−1 −2 −1 −1} {−8 −12 −77 26 −8 7 −2 2 −11 −18 −79 16 −4 4 0 1} {2 11 −61 −1 0 0 0 −3 −37 3 0 1 0 1 0} },

-   -   -   lowFreqTransMatrix[m] [n]=lowFreqTransMatrixCol32to47[m−32]            [n] with m=32 . . . 47, n=0 . . . 15

lowFreaTransMatrixCol32to47 = [[{ {3 −1 0 0 2 −1 0 0 2 −1 0 0 1 0 0 0}{−12 2 1 0 −5 1 0 0 −1 0 0 0 −2 0 0 0} {17 −3 −1 0 6 −1 −1 0 2 0 0 0 2 00 0} {−7 −1 2 0 −3 −1 1 0 −2 −2 1 0 0 0 0 0} {−32 −3 3 0 12 −2 −1 0 7 00 0 1 0 0 0} {−3 −19 3 0 −4 −6 1 0 0 0 0 0 0 −1 0 0} {117 −10 −8 0 32 1−4 0 3 1 −1 0 −3 1 0 0} {−7 32 −5 1 −1 4 0 0 2 −1 0 0 1 0 −1 0} {4 10 5−1 0 3 1 0 −2 1 2 0 −1 1 1 0} {30 13 −3 0 −116 6 10 0 −35 −5 4 0 −3 −1 00} {−10 −63 1 2 −17 3 −4 0 −1 9 −1 0 3 4 −1 0} {2 −3 −4 0 2 −2 −2 0 0 0−1 0 0 −1 −1 0} {−8 −2 −1 1 30 4 −4 1 −102 4 8 −1 −69 −2 6 −1} {1 −95 18−6 −10 −34 −2 0 −4 17 −2 0 0 2 1 0} {2 10 24 −7 5 9 19 −1 0 1 4 0 −2 0 10} {−1 −2 −4 4 0 3 1 −1 0 2 0 −2 2 0 0 0} }, ]] { {−3 1 0 0 −2 1 0 0 −21 0 0 −1 0 0 0} {−15 2 1 0 −5 1 0 0 −1 0 0 0 −1 1 0 0} {−16 4 1 0 −7 2 10 −2 1 0 0 −2 0 0 0} {−15 −1 3 0 −2 −1 1 0 −1 −3 1 0 0 0 0 0} {−38 −3 61 14 −3 −1 0 11 0 −1 0 3 0 0 0} {−7 −24 6 0 −5 −8 2 0 0 0 0 −1 0 −2 0 0}{−111 16 11 1 −38 0 7 0 2 −1 1 0 4 −1 −1 0} {9 −22 6 −2 1 −8 1 0 −4 −1 1−1 −1 0 0 0} {−33 −32 7 2 100 −10 −16 0 46 8 −10 −1 4 3 −1 0} {−5 −66 74 −42 6 1 2 −13 13 0 −1 1 4 −1 0} {1 11 5 −1 14 4 −1 −1 2 2 2 −1 −1 0 10} {−16 −5 −8 3 34 7 −11 0 −81 8 14 0 −44 −1 9 0} {11 −8 −11 2 −27 −16 02 60 −6 −10 0 33 2 −8 −1} {1 −78 23 1 −10 −62 7 6 −16 2 1 1 −4 3 0 0} {0−9 −3 −3 3 5 13 −4 0 7 5 −2 0 3 0 −1} {1 35 4 −5 8 −85 20 4 −11 −72 11 8−6 −10 1 1} },

-   -   Otherwise, if nTrS is equal to 48, lfnstTrSetIdx is equal to 2,        and lfnst_idx is equal to 2 the following applies:        -   lowFreqTransMatrix[m] [n]=lowFreqTransMatrixCol0to15[m] [n]            with m=0 . . . 15, n=0 . . . 15

lowFreqTransMatrixCol0to15 = [[{ {87 −41 3 −4 1 −1 0 −1 −73 28 2 1 1 1 00} {−75 4 7 0 2 0 1 0 −41 36 −7 3 −1 1 0 0} {26 −44 22 −6 4 −2 1 −1 7724 −22 2 −4 0 −1 0} {−39 −68 37 −7 6 −2 2 0 −9 56 −21 1 −2 0 −1 0} {10−20 2 0 1 0 0 0 50 −1 8 −5 1 −1 0 0} {−21 −45 8 −2 3 −1 1 0 −7 −30 26 −83 −1 1 −1} {−4 −2 −55 28 −8 5 −3 2 −2 37 43 −19 1 −2 1 −1} {2 19 47 −236 −4 2 −1 −23 −22 −44 17 −2 2 −1 0} {−19 −62 −9 3 0 0 0 0 −12 −56 27 −73 −1 1 0} {1 9 −5 0 −1 0 0 0 0 22 −1 2 0 1 0 0} {5 17 −9 0 −2 1 0 0 1354 −2 7 −1 1 0 0} {7 27 56 −2 10 −3 3 −1 −2 −6 8 −28 3 −4 1 −1} {0 0 19−4 3 −2 2 −1 −3 −13 10 −4 1 0 0 0} {−3 0 −27 −80 40 −16 6 −4 4 3 31 61−22 7 1} {1 2 −8 6 −1 1 0 0 2 8 −5 −1 0 0 0 0} {−4 −18 −57 8 −8 1 −3 0−5 −20 −69 7 −6 2 −2 1} }, ]] { {80 −43 3 −4 1 −1 0 −1 −75 34 2 1 1 1 00} {73 −8 −11 −1 −3 0 −1 0 26 −35 11 −3 2 −1 1 0} {37 −40 19 −6 3 −1 1−1 73 17 −25 1 −5 0 −2 0} {43 62 −41 6 −8 1 −3 0 5 −60 25 0 3 1 1 0}{−10 26 −5 0 −1 0 −1 0 −55 1 −5 6 0 2 0 1} {−23 −40 10 1 4 0 2 0 −9 −1927 −8 3 −2 1 −1} {−4 14 −12 7 −3 2 −1 1 −23 15 5 −5 0 −1 1 0} {1 5 63−36 10 −8 3 −3 −21 −45 −52 31 −1 5 0 2} {22 61 11 −13 −1 −3 −1 −1 10 38−42 14 −5 2 −1 1} {2 14 −17 5 −2 2 −1 0 3 39 0 1 −4 1 −1 0} {5 19 35 −85 −4 1 −1 −8 −20 1 −14 4 −1 1 −1} {−13 −38 −19 5 −1 2 1 1 −11 −42 4 13−1 3 −1 1} {−3 −2 −31 11 −6 5 −2 2 6 31 −11 10 −2 0 −1 0} {−2 0 14 −2310 −4 2 −1 −3 −10 18 14 −5 1 −1 0} {0 −7 8 65 −31 12 −8 4 −9 −21 −39 −5224 −4 6 0} {8 20 55 26 −15 3 −4 0 2 14 11 −45 14 −4 3 −2} },

-   -   -   lowFreqTransMatrix[m] [n]=lowFreqTransMatrixCol16to31[m−16]            [n] with m=16 . . . 31, n=0 . . . 15

lowFreqTransMatrixCol16to31 = [[{ {30 −5 −6 1 −1 0 0 0 −8 −3 3 0 0 0 00} {72 −29 −2 0 −1 0 −1 0 −37 6 7 −2 1 0 0 0} {7 −38 10 0 1 0 0 0 −51 274 −3 2 −1 1 0} {−45 4 −3 6 −1 2 0 1 49 −13 3 −3 −1 0 0 0} {66 17 −24 4−3 1 −1 0 13 −49 15 1 0 0 0 0} {−9 69 −33 5 −2 0 −1 0 −44 −31 10 7 −2 20 1} {−47 −34 −27 5 4 −1 1 0 −39 −2 27 4 −2 1 0 0} {−33 3 22 −2 −4 1 −10 −58 −17 6 −6 7 −1 1 0} {7 −8 16 −6 4 −2 1 −1 −15 54 −23 2 −1 0 0 0}{−13 17 0 −2 0 −1 0 0 −46 −10 −10 4 −1 1 0 0} {4 51 −3 −6 −1 −1 0 0 −206 −34 9 −2 2 −1 0} {−1 −4 −68 35 −5 5 −2 1 0 35 43 −4 −6 1 −1 0} {−6 −37−18 −5 2 −2 1 −1 6 −6 −7 25 −6 4 −1 1} {−4 −7 −26 −6 −10 6 −4 1 3 8 14−18 15 −5 2 −1} {1 24 3 5 −1 1 0 0 −3 12 6 −10 1 −1 0 0} {1 4 0 33 −7 5−2 1 0 −9 53 −22 3 −1 0 0} }, ]] { {33 −7 −7 1 −1 0 −1 0 −9 −4 5 0 0 0 00} {−70 37 −1 0 1 0 0 0 43 −12 −8 3 −2 1 −1 0} {−11 −38 16 1 2 1 1 0 −4535 1 −4 2 −1 1 0} {41 −3 3 −8 2 −3 0 −1 −50 13 −3 4 1 1 0 0} {−55 −13 29−5 4 −1 2 0 10 52 −23 −2 −2 −1 −1 −1} {−18 63 −36 6 −3 0 −1 0 −45 −22 128 −2 3 0 1} {−67 −27 1 6 3 1 1 1 −54 −3 30 −4 4 −1 1 0} {0 29 33 −8 −6 0−2 0 −33 −7 −16 −4 8 −1 2 0} {−3 5 −4 3 −5 2 −1 1 21 −53 21 −4 5 −1 1 0}{−22 1 −1 −9 4 −2 1 −1 −46 −24 −15 19 −3 4 0 1} {−7 −9 −40 32 −7 5 −2 2−11 15 45 −19 0 −2 −1 −1} {−3 −35 44 −19 6 −3 2 −1 12 −24 −2 3 4 −1 1 0}{3 48 31 −17 1 −3 0 −1 −27 −11 −24 −5 6 −1 2 0} {−7 −37 −16 −6 3 2 1 1−1 −13 −2 16 −2 1 0 0} {1 −11 23 27 −2 −2 0 −1 −4 −35 9 −8 −4 4 0 1} {515 4 −6 8 −7 3 −2 2 35 −41 27 −15 6 −3 1} },

-   -   -   lowFreqTransMatrix[m] [n]=lowFreqTransMatrixCol32to47[m−32]            [n] with m=32 . . . 47, n=0 . . . 15

lowFreqTransMatrixCol32to47 = [[{ {3 2 −1 0 −2 −1 0 0 1 1 0 0 −1 0 0 0}{12 3 −4 0 −3 −2 1 0 4 0 0 0 −1 0 0 0} {31 −5 −8 3 −14 0 5 −1 6 1 −3 0−4 −1 1 0} {−19 2 0 0 5 1 1 0 −2 0 −1 0 1 0 0 0} {−53 34 6 −5 30 −7 −113 −11 −2 5 1 4 2 −1 −1} {49 7 2 −6 −23 −3 −2 2 9 4 0 0 −2 −1 −1 0} {−1132 −8 −7 27 −12 −6 6 −13 0 4 −3 3 −1 −2 1} {−23 40 −2 5 43 −11 −8 −1 −18−4 5 2 4 3 0 −1} {−42 −25 4 6 34 8 2 −2 −15 −1 0 −1 3 2 0 1} {−80 −27 20−4 −66 23 −2 −2 20 −3 −2 3 −14 2 3 −1} {16 −52 28 1 59 15 −8 −5 −28 −7 22 10 3 0 −1} {−14 −38 −12 −10 9 5 7 6 −9 7 −4 −3 4 −4 0 3} {16 10 55 −2415 46 −52 1 35 −43 10 12 −23 13 5 −8} {−2 −4 −1 13 0 2 −4 −3 3 −1 2 1 −20 −2 −1} {−9 −1 −25 10 45 −11 18 2 86 1 −13 −4 −65 −6 7 2} {4 −27 −2 −95 36 −13 5 −7 −17 1 2 4 6 4 −1} }, ]] { {3 3 −1 0 −2 −2 0 0 1 1 0 0 −1 00 0} {−17 −2 7 −1 5 3 −3 0 −5 0 1 0 1 1 −1 0} {31 −11 −10 4 −16 0 8 −2 52 −4 0 −3 −1 2 0} {18 0 0 0 −4 −2 −1 0 0 0 1 0 0 0 −1 0} {43 −43 −3 7−27 10 14 −5 11 3 −8 0 −2 −3 3 1} {59 1 1 −9 −29 −3 0 3 9 6 −1 −1 −1 −2−1 0} {−11 53 −19 −2 40 −26 −4 6 −17 2 8 −4 6 1 −4 1} {−15 7 11 9 21 8−7 −7 −6 −9 2 4 2 6 1 −2} {45 24 −10 −5 −38 −14 1 3 11 2 2 0 −1 −3 −1 0}{−39 −15 51 −13 −26 51 −29 −4 26 −27 −2 11 −12 6 8 −6} {−58 −18 −23 4−44 9 28 6 26 17 −15 −7 −8 −11 3 6} {−11 52 −5 −1 −59 −6 0 4 37 4 1 0−11 0 1 −2} {−38 −15 −14 23 −23 −35 43 −8 −19 42 −10 −7 15 −11 −5 5} {−5−18 29 −13 −44 −11 −11 4 −76 20 27 2 43 8 −16 −3} {10 −14 2 5 −5 27 8 −7−33 −7 −6 5 18 4 6 0} {−11 34 13 −20 −21 −44 −13 12 9 2 15 −1 −8 5 −6−3} },

-   -   Otherwise, if nTrS is equal to 48, lfnstTrSetIdx is equal to 3,        and lfnst_idx is equal to 1 the following applies:        -   lowFreqTransMatrix[m] [n]=lowFreqTransMatrixCol0to15[m] [n]            with m=0 . . . 15, n=0 . . . 15

lowFreqTransMatrixCol0to15 = [{ {−115 37 9 2 2 1 1 0 10 −29 8 0 1 0 1 0}{15 51 −18 0 −3 0 −1 0 −95 7 34 −3 5 −1 2 0} {29 −22 16 −6 3 −2 1 −1 −4−80 12 15 0 3 0 1} {−36 −98 25 5 4 1 2 1 −59 11 −17 1 1 1 0 0} {−6 18 3−3 −1 0 0 0 −50 −5 −38 12 0 2 0 1} {4 15 52 −13 5 −3 2 −1 −17 −45 16 24−2 4 −1 2} {−20 −7 −43 4 0 1 −1 1 −7 35 0 12 −4 1 −1 0} {4 29 1 26 −5 4−2 1 −17 −7 −73 6 6 2 1 1} {12 13 10 2 −1 3 −1 1 17 −2 −46 12 7 0 2 0}{5 20 90 −17 4 −3 2 −1 6 66 8 28 −7 3 −1 1} {−3 −4 −34 −12 2 −1 −1 0 525 11 43 −10 4 −2 1} {−1 −3 2 19 −2 4 −1 2 9 3 −35 22 11 1 2 0} {10 −4−6 12 5 1 1 0 11 −9 −12 −2 −7 0 −1 0} {4 6 14 53 −4 4 0 2 0 −1 −20 −13 32 −1 1} {2 9 13 37 19 6 2 2 −9 −3 −9 −28 −20 −4 −3 −1} {3 −3 12 84 −12 8−2 3 6 13 50 −1 45 1 7 0} }, ]] { {112 −38 −11 −2 −3 −1 −1 0 −17 33 −7−1 −1 0 −1 0} {7 48 −20 0 −3 0 −1 0 −86 15 39 −6 5 −1 2 0} {−34 14 −7 5−2 2 0 1 7 74 −19 −16 0 −4 0 −1} {−17 −69 19 11 3 2 1 1 14 7 5 −8 1 −1 00} {31 66 −29 −3 −4 −2 −2 −1 78 −1 30 −13 −3 −3 −1 −1} {−24 −12 −48 15 14 0 2 −2 57 −19 −8 −2 −2 −1 −1} {−8 0 29 −14 6 −3 2 −1 −13 −15 29 27 −84 −2 1} {5 39 5 11 −6 0 −2 0 −30 −14 −58 19 10 4 2 2} {−13 −16 7 2 3 −32 −1 −18 13 46 −14 −8 1 −3 0} {−12 −15 −67 16 5 4 0 2 0 5 0 22 −3 0 −10} {−5 −22 −68 21 1 4 1 2 −13 −67 3 −44 11 1 4 1} {−11 0 5 −12 −8 0 0 0−14 −1 13 −7 7 −2 0 0} {−5 −10 −17 −51 10 −4 3 −1 1 14 50 −8 −18 −1 −4−1} {−2 −16 −6 −19 2 2 1 1 13 −4 11 11 25 −4 3 −1} {−7 2 2 −5 7 5 0 1−14 0 2 −7 −13 0 1 0} {6 7 31 −8 29 −7 3 −2 −3 −10 −6 −71 14 0 4 0} },

-   -   -   lowFreqTransMatrix[m] [n]=lowFreqTransMatrixCol16to31[m−16]            [n] with m=16 . . . 31, n=0 . . . 15

lowFreqTransMatrixCol16to31 = [[{ {23 −8 −8 1 −1 0 0 0 3 3 −2 −1 0 0 00} {23 −47 1 6 0 1 0 1 8 5 −12 0 −1 0 0 0} {45 7 −59 7 −2 1 −1 0 −15 41−3 −16 2 −3 0 −1} {6 −13 7 −3 0 0 0 0 14 −4 −14 3 −1 0 0 0} {3 67 −7 −403 −6 1 −3 −12 −13 65 −3 −10 0 −1 0} {−87 −8 −14 7 8 1 2 0 23 −35 −6 −3 11 0 0 } {−51 −2 −57 5 15 0 4 0 7 39 5 −55 1 −7 1 −3} {−5 21 −3 5 −1 −3 0−1 −11 2 −52 −3 27 −2 5 0} {16 −45 −9 −53 6 1 1 0 70 16 8 −4 −37 1 −7 0}{29 5 −19 12 9 −1 1 0 −10 14 −1 −13 7 0 1 0} {23 20 −40 12 21 −3 4 −1 25−28 −10 5 8 6 0 2} {−7 −65 −19 −22 11 4 2 1 −75 −18 3 −1 −10 2 0 1} {33−10 −4 18 18 −4 4 −1 28 −72 1 −49 15 2 2 1} {−3 1 −5 35 −16 −6 −1 −2 4629 13 21 37 −5 4 −1} {1 18 9 28 24 6 2 2 −20 −5 −25 −33 −36 9 −2 2} {−218 −22 −37 −13 14 0 3 1 −12 −3 2 −15 −8 1 −1} }, ]] { {−23 7 10 −1 1 0 00 −2 −4 2 2 0 0 0 0} {22 −56 8 11 0 2 0 1 11 3 −19 2 0 0 0 0} {−38 −2 65−11 0 −2 1 −1 14 −47 7 22 −3 3 −1 1} {0 −49 12 33 −4 5 −1 1 9 6 −69 8 130 2 0} {−19 −21 5 29 −2 3 −1 1 −12 8 −26 −1 10 −1 2 0} {−11 2 −36 1 11−1 3 0 0 44 2 −54 4 −6 2 −2} {−93 −13 −32 15 20 2 5 1 29 −23 2 −30 −2 00 0} {−23 34 −7 0 1 −5 1 −2 −14 −13 −37 −2 31 −1 5 0} {−3 46 6 54 −11 −5−2 −2 −57 −13 6 4 42 −3 7 −1} {−4 2 −29 7 16 −1 3 0 −6 −8 −3 17 4 3 0 0}{−31 8 31 −6 −17 1 −2 0 13 −16 1 −4 −2 −2 0 0} {−47 10 6 −15 −37 7 −3 2−4 75 5 56 −10 −20 −1 −5} {17 42 −2 −6 3 −3 −1 −1 32 −12 1 −8 −7 4 −2 1}{−9 −48 −1 −50 13 12 2 3 −50 −16 20 −7 −6 3 1 1} {−16 6 7 −6 14 14 −2 3−82 1 −26 −10 −54 12 2 3} {0 0 15 −9 17 −4 −2 0 5 6 4 −36 10 13 −1 3} },

-   -   -   lowFreqTransMatrix[m] [n]=lowFreqTransMatrixCol32to47[m−32]            [n] with m=32 . . . 47, n=0 . . . 15

lowFreqTransMatrixCol32to47 = [[{ {4 0 0 −1 1 1 0 0 2 0 0 0 0 0 0 0} {3−3 1 −1 2 1 −2 0 1 −1 0 0 1 1 −1 0} {1 0 7 −2 −3 6 1 −2 0 0 1 0 −1 2 0−1} {2 8 −3 −5 2 0 0 0 0 3 0 −1 1 0 0 0} {9 −20 −5 22 −2 0 0 −1 2 −3 −23 −1 0 1 0} {2 5 −17 0 3 −1 −1 −5 0 1 −4 0 1 0 0 −2} {1 −10 41 2 4 −3 −23 −1 −2 7 1 1 −1 −1 0} {0 27 8 −58 2 −5 25 3 0 3 0 −5 0 −2 7 0} {−12 293 21 4 0 5 −1 −3 4 1 4 2 0 1 0} {0 −6 13 −4 0 −4 1 5 0 −1 −1 1 0 −1 0 0}{−4 21 −64 −8 −5 19 10 −48 3 −1 10 −3 0 4 3 −6} {2 −35 −27 4 1 8 −17 −193 0 3 −6 0 2 −1 −2} {56 −23 22 −1 4 −1 −15 26 6 4 −10 0 0 2 −3 2} {−10−53 −18 8 9 12 −41 −25 −2 2 13 −16 4 1 −5 1} {−13 42 1 57 −22 −2 −25 −285 6 19 −12 −5 −3 −2 4} {19 14 −4 −12 −4 5 17 8 2 −4 −4 4 −2 2 1 0} }, ]]{ {−4 0 0 1 −1 −1 0 0 −2 0 0 0 0 0 0 0} {3 −1 1 −3 3 1 −2 0 1 −1 0 0 1 0−1 0} {0 0 −11 4 3 −6 −1 2 0 0 −2 0 1 −2 0 0} {−5 25 2 −32 3 0 5 1 −1 51 −3 1 0 1 1} {−7 3 8 −11 −2 −2 1 1 −2 0 2 0 −1 0 0 1} {4 −10 53 −4 3 −60 9 0 −2 7 1 1 −2 0 2} {6 −1 −1 13 4 −1 −8 −5 1 1 0 0 2 0 −2 −1} {6 17 2−62 6 −6 31 2 2 0 −2 −2 1 −1 6 0} {15 −43 0 −12 −2 1 −15 4 4 −4 −2 −6 01 −2 0} {−14 0 −65 −2 0 17 4 −64 5 −1 18 −2 0 2 2 −3} {19 5 15 4 −1 4 −523 1 3 −4 −2 0 1 −1 2} {−35 −9 2 −9 12 −11 14 −12 −2 −5 3 10 2 −3 −1 −1}{5 41 0 −27 3 −2 56 6 −2 −6 −5 32 −2 0 −1 0} {18 −43 −3 −44 20 8 25 10 00 −9 26 1 3 −3 −3} {2 52 5 33 −14 −11 8 −6 7 −2 3 −4 −1 −2 3 1} {−15 −121 −11 8 −10 4 −63 13 1 39 6 0 −1 −2 18} },

-   -   Otherwise, if nTrS is equal to 48, lfnstTrSetIdx is equal to 3,        and lfnst_idx is equal to 2 the following applies:        -   lowFreqTransMatrix[m] [n]=lowFreqTransMatrixCol0to15[m] [n]            with m=0 . . . 15, n=0 . . . 15

lowFreaTransMatrixCol0to15 = [[{ {109 −26 −8 −3 −2 −1 −1 0 −50 28 2 1 00 0 0} {−39 31 −5 2 −1 1 0 0 −95 6 18 0 4 0 1 0} {29 −3 −2 −2 0 0 0 0 0−41 9 0 2 0 1 0} {18 96 −23 2 −5 1 −2 0 −10 6 10 −2 1 −1 1 0} {−29 −6016 −2 3 −1 1 0 −52 9 −17 5 −2 1 −1 1} {−23 −5 −15 5 −2 1 −1 1 2 79 −13−4 −2 −1 −1 0} {−7 −3 12 −3 3 −1 1 0 −31 −62 8 7 0 2 0 1} {1 −26 5 0 1 01 0 24 −3 43 −6 4 −2 1 −1} {11 14 6 −3 1 −1 1 0 10 −7 −9 3 −2 1 −1 0}{−10 −11 −47 3 −4 1 −1 0 5 28 11 −2 −1 0 0 0} {−8 −24 −99 11 −10 3 −4 1−5 −36 19 −26 4 −5 1 −2} {−5 1 −1 0 1 0 0 0 −10 −14 −6 8 0 1 0 0} {1 12−20 21 −4 5 −2 2 −5 −2 −75 9 −1 2 −1 1} {2 −9 −18 8 −3 3 −1 1 3 −25 −62−6 0 −2 0 −1} {4 9 39 18 0 2 0 1 −6 −16 −22 −37 5 −5 1 −2} {−7 −2 15 −61 −1 1 −1 −11 −3 22 −14 0 −2 1 −1} }, ]] { {−111 32 9 3 2 1 1 0 39 −30 1−1 0 0 0 0} {25 −26 6 −2 1 −1 0 0 94 −12 −20 0 −4 0 −2 0} {29 −2 −5 −1−1 0 0 0 −1 −35 10 0 2 0 1 0} {3 54 −15 −1 −4 −1 −1 0 −36 9 10 0 2 0 10} {−9 38 −20 4 −4 1 −1 0 18 59 −5 −10 −2 −3 −1 −1} {−42 −87 26 4 7 2 31 −34 37 −22 3 −2 1 −1 0} {7 −20 3 3 1 1 0 0 33 −12 34 −7 2 −2 1 −1} {53 −12 4 −3 1 −1 0 32 66 −14 −11 −2 −4 −1 −1} {−15 −11 −32 6 1 2 0 1 1 356 −6 −1 −1 −1 −1} {9 16 −9 −3 −1 −1 0 0 13 4 −8 2 −4 1 −1 0} {−3 11 20−3 3 −1 1 0 −16 −15 −12 19 0 4 0 1} {2 18 −17 13 −4 2 −2 1 −9 4 −58 7 22 0 1} {14 32 101 −22 3 −7 0 −2 9 33 −13 22 −5 3 −2 1} {−8 −2 7 −4 2 −21 0 −4 28 72 0 −7 −1 −2 −1} {3 4 35 −1 −2 −2 0 −1 −11 −26 10 −38 8 −3 30} {−6 −7 −5 10 −1 2 0 1 −9 −12 −28 6 3 1 1 1} },

-   -   -   lowFreqTransMatrix[m] [n]=lowFreqTransMatrixCol16to31[m−16]            [n] with m=16 . . . 31, n=0 . . . 15

lowFreqTransMatrixCol16to31 = [[{ {−18 −8 6 0 1 0 1 0 6 −2 −3 0 0 0 0 0}{32 −49 5 1 1 0 0 0 27 −1 −14 2 −2 1 −1 0} {86 4 −33 2 −6 1 −2 0 −32 581 −7 0 −2 0 −1} {−14 26 2 −4 1 −1 0 0 −43 −9 35 −2 4 −1 1 0} {13 56 −2−9 0 −2 0 −1 −34 −18 41 0 3 0 1 0} {−9 1 5 −1 1 0 0 0 −4 49 2 −14 1 −3 0−1} {−75 9 −45 5 −1 1 −1 0 14 35 0 −23 2 −5 1 −2} {−7 −64 9 14 0 3 0 1−12 −4 5 3 −1 1 0 0} {22 21 1 −21 2 −4 1 −2 92 1 53 0 −9 1 −2 0} {−12 −2−38 2 0 1 0 0 16 38 11 −16 −1 −3 0 −2} {0 25 41 5 −3 1 0 0 10 −5 −7 12 21 0 0} {−17 −2 7 −5 3 −1 0 0 −16 13 3 31 −1 6 0 2} {−1 −2 −16 −4 0 −1 00 −7 7 −31 0 3 0 0 0} {−6 −61 14 −51 2 −6 0 −2 −19 0 40 −7 −17 0 −3 0}{−5 15 63 9 −16 0 −3 0 18 42 −18 27 15 1 3 1} {−18 −7 30 −9 −4 0 −1 0−35 23 23 10 −17 1 −3 0} }, ]] { {23 2 −7 0 −1 0 −1 0 −4 3 1 0 0 0 0 0}{−31 56 −7 −2 −1 −1 0 0 −30 3 19 −3 2 −1 1 0} {77 −1 −36 3 −6 1 −2 0 −2967 −3 −11 0 −3 0 −1} {−4 37 −1 −7 0 −2 0 −1 −46 −12 52 −3 5 −1 2 0} {−34−15 13 4 2 1 1 0 5 40 −11 −15 1 −3 0 −1} {11 24 −2 −6 0 −2 0 −1 0 11 18−8 0 −2 0 −1} {−11 −65 8 20 0 5 0 2 3 1 −9 3 −1 1 0 0} {70 −12 45 −7 −1−2 −1 −1 −25 −20 3 28 −2 5 −1 1} {−23 −10 −37 9 3 3 1 1 −11 39 −5 −21 1−4 1 −1} {18 15 −13 −21 3 −4 1 −1 90 10 49 −12 −14 −2 −4 −1} {−25 4 −4−3 4 0 1 0 −12 17 −4 29 −2 4 −1 1} {6 23 −13 4 −1 −1 0 0 1 1 −38 −6 11−1 2 −1} {6 −21 −34 3 6 2 2 0 −8 −6 2 −18 1 0 0 0} {−1 41 −16 48 −5 3 −21 −5 −1 −16 9 8 1 1 0} {−4 21 65 −1 −17 −1 −4 0 4 16 −7 10 2 −1 0 0}{−21 −42 8 −24 7 2 2 0 −49 14 29 −5 −25 3 −4 1} },

-   -   -   lowFreqTransMatrix[m] [n]=lowFreqTransMatrixCol32to47[m−32]            [n] with m=32 . . . 47, n=0 . . . 15

lowFreqTransMatrixCol32to47 = [[{ {−3 2 1 −1 0 0 0 0 −2 0 0 0 0 0 0 0}{3 5 −3 −2 4 1 −1 −1 2 0 0 0 2 0 0 0} {−14 −8 20 0 −2 −3 0 4 −1 −1 0 0−1 1 0 0} {14 −40 1 10 2 1 −10 1 2 −4 −1 −1 0 0 −1 0} {19 −36 −10 13 3 6−14 −1 3 1 −1 −3 1 1 −1 −1} {−31 −14 56 −1 13 −37 −4 20 −2 2 −10 0 2 −40 −1} {1 −8 32 −1 7 −12 −4 10 0 2 −6 −1 2 0 0 −2} {8 −59 −3 26 14 6 −586 −5 17 −7 −18 3 3 −1 −5} {−21 −11 1 40 −5 −4 −24 5 −4 5 −6 −5 0 0 0 −3}{12 −9 −22 7 −8 60 4 −36 −6 −15 54 7 3 −7 −8 14} {−1 1 9 −3 −3 −14 −3 122 4 −13 −2 −1 3 2 −4} {−93 −15 −46 −3 23 −19 0 −47 8 4 8 3 2 3 0 0} {411 −12 4 −12 14 −50 −1 −8 32 −4 −54 2 0 30 −15} {13 −4 11 9 17 0 24 5 1−12 4 28 0 0 −15 8} {12 −34 9 −24 4 28 −2 4 −11 4 30 2 5 −13 −4 18} {−1953 6 48 −65 12 −12 11 −8 16 10 −21 −2 −12 6 2} }, ]] { {3 −2 −1 0 0 1 −1−1 2 0 0 0 0 0 0 0} {−2 −9 4 3 −4 −1 0 1 −1 0 0 0 −1 0 0 0} {−17 −7 29−2 −1 −8 1 8 −2 −1 −2 1 −1 0 0 0} {19 −62 1 20 5 3 −25 3 2 0 −2 −6 2 1 0−1} {−23 −2 61 −7 9 −41 3 25 −3 0 −14 3 1 −4 −1 −2} {−3 −23 21 10 8 −13−18 11 −1 7 −7 −5 2 −1 1 −3} {2 −45 −1 27 7 9 −65 6 −6 24 −5 −26 3 2 5−6} {−20 0 −28 4 −3 4 0 −14 −1 −1 7 −1 −2 −1 0 3} {3 −11 −17 −4 5 47 6−48 2 −11 61 5 2 −9 −6 21} {−24 3 0 45 −16 8 −18 −6 −4 −3 11 −4 −2 −3 −23} {−86 −8 −33 1 23 −35 −11 −44 11 12 0 −11 4 3 8 −4} {14 22 1 −10 −1913 −41 2 −3 30 5 −62 6 −5 41 −6} {11 2 4 1 −2 11 1 1 −3 −1 13 −1 0 −6 28} {−15 30 −9 22 −48 −8 −11 2 1 −23 −11 −24 1 −1 1 −13} {−2 1 30 1 −12−1 −10 −5 9 −5 53 −14 2 −21 2 50} {2 46 1 51 −43 −5 −4 5 9 −33 −1 −16 1−1 −11 −2} },

FIG. 23A is a block diagram of a video processing apparatus 2300. Theapparatus 2310 may be used to implement one or more of the methodsdescribed herein. The apparatus 2310 may be embodied in a smartphone,tablet, computer, Internet of Things (IoT) receiver, and so on. Theapparatus 2310 may include one or more processors 2312, one or morememories 2304 and video processing hardware 2316. The processor(s) 2312may be configured to implement one or more methods (including, but notlimited to, methods 2200, 2210, 2220, 2230, 2240 and 2250) described inthe present document. The memory (memories) 2304 may be used for storingdata and code used for implementing the methods and techniques describedherein. The video processing hardware 2306 may be used to implement, inhardware circuitry, some techniques described in the present document.In some embodiments, the hardware 2306 may be at least partly internalto the processor 2312, e.g., a graphics co-processor.

FIG. 23B is another example of a block diagram of a video processingsystem in which disclosed techniques may be implemented. FIG. 23B is ablock diagram showing an example video processing system 2320 in whichvarious techniques disclosed herein may be implemented. Variousimplementations may include some or all of the components of the system2320. The system 2320 may include input 2322 for receiving videocontent. The video content may be received in a raw or uncompressedformat, e.g., 8 or 10 bit multi-component pixel values, or may be in acompressed or encoded format. The input 2322 may represent a networkinterface, a peripheral bus interface, or a storage interface. Examplesof network interface include wired interfaces such as Ethernet, passiveoptical network (PON), etc. and wireless interfaces such as Wi-Fi orcellular interfaces.

The system 2320 may include a coding component 2324 that may implementthe various coding or encoding methods described in the presentdocument. The coding component 2324 may reduce the average bitrate ofvideo from the input 2322 to the output of the coding component 2324 toproduce a coded representation of the video. The coding techniques aretherefore sometimes called video compression or video transcodingtechniques. The output of the coding component 2324 may be eitherstored, or transmitted via a communication connected, as represented bythe component 2326. The stored or communicated bitstream (or coded)representation of the video received at the input 2322 may be used bythe component 2328 for generating pixel values or displayable video thatis sent to a display interface 2340. The process of generatinguser-viewable video from the bitstream representation is sometimescalled video decompression. Furthermore, while certain video processingoperations are referred to as “coding” operations or tools, it will beappreciated that the coding tools or operations are used at an encoderand corresponding decoding tools or operations that reverse the resultsof the coding will be performed by a decoder.

Examples of a peripheral bus interface or a display interface mayinclude universal serial bus (USB) or high definition multimediainterface (HDMI) or Display port, and so on. Examples of storageinterfaces include SATA (serial advanced technology attachment), PCI,IDE interface, and the like. The techniques described in the presentdocument may be embodied in various electronic devices such as mobilephones, laptops, smartphones or other devices that are capable ofperforming digital data processing and/or video display.

Some embodiments of the disclosed technology include making a decisionor determination to enable a video processing tool or mode. In anexample, when the video processing tool or mode is enabled, the encoderwill use or implement the tool or mode in the processing of a block ofvideo, but may not necessarily modify the resulting bitstream based onthe usage of the tool or mode. That is, a conversion from the block ofvideo to the bitstream representation of the video will use the videoprocessing tool or mode when it is enabled based on the decision ordetermination. In another example, when the video processing tool ormode is enabled, the decoder will process the bitstream with theknowledge that the bitstream has been modified based on the videoprocessing tool or mode. That is, a conversion from the bitstreamrepresentation of the video to the block of video will be performedusing the video processing tool or mode that was enabled based on thedecision or determination.

Some embodiments of the disclosed technology include making a decisionor determination to disable a video processing tool or mode. In anexample, when the video processing tool or mode is disabled, the encoderwill not use the tool or mode in the conversion of the block of video tothe bitstream representation of the video. In another example, when thevideo processing tool or mode is disabled, the decoder will process thebitstream with the knowledge that the bitstream has not been modifiedusing the video processing tool or mode that was disabled based on thedecision or determination.

In the present document, the term “video processing” may refer to videoencoding video decoding, video compression or video decompression. Forexample, video compression algorithms may be applied during conversionfrom pixel representation of a video to a corresponding bitstreamrepresentation or vice versa. The bitstream representation of a currentvideo block may, for example, correspond to bits that are eitherco-located or spread in different places within the bitstream, as isdefined by the syntax. For example, a macroblock may be encoded in termsof transformed and coded error residual values and also using bits inheaders and other fields in the bitstream.

In some embodiments, the video coding methods may be implemented usingan apparatus that is implemented on a hardware platform as describedwith respect to FIG. 23A or 23B.

Various techniques and embodiments may be described using the followingclause-based format. The first set of clauses describe certain featuresand aspects of the disclosed techniques in the previous section.

1. A method for video processing, comprising: selecting, based on acharacteristic of a current video block, a transform set or a transformmatrix for an application of a reduced secondary transform to thecurrent video block; and applying, as part of a conversion between thecurrent video block and a bitstream representation of a video comprisingthe current video block, the selected transform set or transform matrixto a portion of the current video block.

2. The method of clause 1, wherein the portion of the current videoblock is a top-right sub-region, bottom-right sub-region, bottom-leftsub-region or center sub-region of the current video block.

3. The method of clause 1 or 2, wherein the characteristic of thecurrent video block is an intra prediction mode or a primary transformmatrix of the current video block.

4. The method of clause 1, wherein the characteristic is a colorcomponent of the current video block.

5. The method of clause 4, wherein a first transform set is selected fora luma component of the current video block, and wherein a secondtransform set different from the first transform set is selected for oneor more chroma components of the current video block.

6. The method of clause 1, wherein the characteristic is an intraprediction mode or an intra coding method of the current video block.

7. The method of clause 6, wherein the intra prediction method comprisesa multiple reference line (MRL)-based prediction method or amatrix-based intra prediction method.

8. The method of clause 6, wherein a first transform set is selectedwhen the current video block is a cross-component linear model (CCLM)coded block, and wherein a second transform set different from the firsttransform set is selected when the current video block is a non-CCLMcoded block.

9. The method of clause 6, wherein a first transform set is selectedwhen the current video block is coded with a joint chroma residualcoding method, and wherein a second transform set different from thefirst transform set is selected when the current video block is notcoded with the joint chroma residual coding method.

10. The method of clause 1, wherein the characteristic is a primarytransform of the current video block.

11. A method for video processing, comprising: making a decision, basedon one or more coefficients associated with a current video block,regarding a selective inclusion of signaling of side information for anapplication of a reduced secondary transform (RST) in a bitstreamrepresentation of the current video block; and performing, based on thedecision, a conversion between the current video block and a videocomprising the bitstream representation of the current video block.

12. The method of clause 11, wherein the one or more coefficientscomprises a last non-zero coefficient in a scanning order of the currentvideo block.

13. The method of clause 11, wherein the one or more coefficientscomprises a plurality of coefficients within a partial region of thecurrent video block.

14. The method of clause 13, wherein the partial region comprises one ormore coding groups that the RST could be applied to.

15. The method of clause 13, wherein the partial region comprises afirst M coding groups or a last M coding groups in a scanning order ofthe current video block.

16. The method of clause 13, wherein the partial region comprises afirst M coding groups or a last M coding groups in a reverse scanningorder of the current video block.

17. The method of clause 13, wherein making the decision is furtherbased on an energy of one or more non-zero coefficients of the pluralityof coefficients.

18. A method for video processing, comprising: configuring, for anapplication of a reduced secondary transform (RST) to a current videoblock, a bitstream representation of the current video block, wherein asyntax element related to the RST is signaled in the bitstreamrepresentation before coding residual information; and performing, basedon the configuring, a conversion between the current video block and thebitstream representation of the current video block.

19. The method of clause 18, wherein signaling the syntax elementrelated to the RST is based on at least one coded block flag or a usageof a transform selection mode.

20. The method of clause 18, wherein the bitstream representationexcludes the coding residual information corresponding to coding groupswith all zero coefficients.

21. The method of clause 18, wherein the coding residual information isbased on the application of the RST.

22. A method for video processing, comprising: configuring, for anapplication of a reduced secondary transform (RST) to a current videoblock, a bitstream representation of the current video block, wherein asyntax element related to the RST is signaled in the bitstreamrepresentation before either a transform skip indication or a multipletransform set (MTS) index; and performing, based on the configuring, aconversion between the current video block and the bitstreamrepresentation of the current video block.

23. The method of clause 22, wherein the transform skip indication orthe MTS index is based on the syntax element related to the RST.

24. A method for video processing, comprising: configuring, based on acharacteristic of a current video block, a context model for coding anindex of a reduced secondary transform (RST); and performing, based onthe configuring, a conversion between the current video block and abitstream representation of a video comprising the current video block.

25. The method of clause 24, wherein the characteristic is an explicitor implicit enablement of a multiple transform selection (MTS) process.

26. The method of clause 24, wherein the characteristic is an enablementof a cross-component linear model (CCLM) coding mode in the currentvideo block.

27. The method of clause 24, wherein the characteristic is a size of thecurrent video block.

28. The method of clause 24, wherein the characteristic is a splittingdepth of a partitioning process applied to the current video block.

29. The method of clause 28, wherein the partitioning process is aquadtree (QT) partitioning process, a binary tree (BT) partitioningprocess or a ternary tree (TT) partitioning process.

30. The method of clause 24, wherein the characteristic is a colorformat or a color component of the current video block.

31. The method of clause 24, wherein the characteristic excludes anintra prediction mode of the current video block and an index of amultiple transform selection (MTS) process.

32. A method for video processing, comprising: making a decision, basedon a characteristic of a current video block, regarding a selectiveapplication of an inverse reduced secondary transform (RST) process onthe current video block; and performing, based on the decision, aconversion between the current video block and a bitstreamrepresentation of a video comprising the current video block.

33. The method of clause 32, wherein the characteristic is a coded blockflag of a coding group of the current video block.

34. The method of clause 33, wherein the inverse RST process is notapplied, and wherein the coded block flag of a top-left coding group iszero.

35. The method of clause 33, wherein the inverse RST process is notapplied, and wherein coded block flags for a first and a second codinggroup in a scanning order of the current video block are zero.

36. The method of clause 32, wherein the characteristic is a height (M)or a width (N) of the current video block.

37. The method of clause 36, wherein the inverse RST process is notapplied, and wherein (i)M=8 and N=4, or (ii) M=4 and N=8.

38. A method for video processing, comprising: making a decision, basedon a characteristic of a current video block, regarding a selectiveapplication of an inverse reduced secondary transform (RST) process onthe current video block; and performing, based on the decision, aconversion between the current video block and a bitstreamrepresentation of a video comprising the current video block; whereinthe bitstream representation includes side information about RST,wherein the side information is included based on coefficients of asingle color or luma component of the current video block.

39. The method of clause 38, wherein the side information is includedfurther based on dimensions of the current video block.

40. The method of any of clauses 38 or 39, wherein the side informationis included without considering block information for the current videoblock.

41. A method of video processing, comprising: determining, for aconversion between a coded representation of a current video blockcomprising sub-blocks and the current video block, a zero-out regionapplied for the conversion of the sub-blocks based on a codingcondition; and performing the conversion based on the determining.

42. The method of clause 41, wherein the coding condition comprises asize of the sub-blocks.

43. The method of any of clauses 41-42, wherein the coding conditionincludes a size of a secondary transform used during the conversion.

44. The method of any of clauses 1 to 43, wherein the conversionincludes generating the bitstream representation from the current videoblock.

45. The method of any of clauses 1 to 43, wherein the conversionincludes generating the current video block from the bitstreamrepresentation.

46. An apparatus in a video system comprising a processor and anon-transitory memory with instructions thereon, wherein theinstructions up on execution by the processor, cause the processor toimplement the method in any one of clauses 1 to 45.

47. A computer program product stored on a non-transitory computerreadable media, the computer program product including program code forcarrying out the method in any one of clauses 1 to 45.

48. A method, apparatus or system described herein.

The second set of clauses describe certain features and aspects of thedisclosed techniques in the previous section (e.g., example items 7-9,22, and 29-34).

1. A method for video processing (e.g., method 2710 shown in FIG. 27A),comprising performing (2712) a conversion between a video region of avideo and a coded representation of the video, wherein the performing ofthe conversion includes configuring, based on a partition type of thevideo region, a context model for coding a first bin, the first bin anda second bin included in a bin string corresponding to an index of asecondary transform tool applied to the video region, wherein the indexindicates an applicability of the secondary transform tool and/or akernel information of the secondary transform tool, and wherein thesecondary transform tool includes applying, during encoding, a forwardsecondary transform to an output of a forward primary transform appliedto a residual of a video block prior to quantization, or wherein thesecondary transform tool includes applying, during decoding, an inversesecondary transform to an output of dequantization to the video blockbefore applying an inverse primary transform.

2. The method of clause 1, wherein the partition type is a single treetype or a dual tree type.

3. The method of clause 1, wherein the conversion is performed for agiven slice type such that the context model used for the given slicetype is different from one used for another slice type.

4. A method for video processing (e.g., method 2720 shown in FIG. 27B),comprising determining (2722), for a current video block of a videocomprising sub-blocks, based on a coding condition of the current videoblock, a zero-out region in which coefficients are zeroed out; andperforming (2724) a conversion between the current video block and acoded representation of the video based on the determining, wherein theconversion includes applying a secondary transform tool, and wherein thesecondary transform tool includes applying, during encoding, a forwardsecondary transform to an output of a forward primary transform appliedto a residual of a video block prior to quantization, or wherein thesecondary transform tool includes applying, during decoding, an inversesecondary transform to an output of dequantization to the video blockbefore applying an inverse primary transform.

5. The method of clause 4, wherein the coding condition comprises a sizeof the sub-blocks.

6. The method of clause 4, wherein nonZeroSizeA indicates a number ofsamples in the zero-out region in case that the size of a currentsub-block is M×N and nonZeroSizeB indicates a number of samples in thezero-out region in case that the size of a current sub-block is K×L, andwherein in case that Min (M, N) is no smaller than Min (K, L),nonZeroSizeA is no smaller than nonZeroSizeB.

7. The method of clause 4, wherein in case that a width or a height of asub-block is equal to 4 or 8, a number of samples in the zero-out regionis set to a fixed value.

8. The method of clause 4, wherein a number of samples in the zero-outregion is set to a fixed value regardless of a size of a sub-block.

9. The method of clause 4, wherein the coding condition includes a sizeof the secondary transform tool.

10. The method of clause 9, wherein coding condition includes whether a16×64 transform or 16×16 transform is used during the conversion.

11. The method of clause 10, wherein nonZeroSizeA indicates a number ofsamples in the zero-out region in case that the 16×64 transform is usedand nonZeroSizeB indicates a number of samples in the zero-out region incase that the 16×16 transform is used, and wherein NonZeroSizeA is nosmaller than NonZeroSizeB.

12. A method for video processing (e.g., method 2730 shown in FIG. 27C),comprising determining (2732), for a conversion between a current videoblock having a dimension N×N of a video and a coded representation ofthe video, to use a transform matrix and/or an inverse transform matrixwith a reduced size that is smaller than N×N in an application of asecondary transform tool to the current video block of a video; andperforming (2734) a conversion between the video and a codedrepresentation of the video, wherein the secondary transform toolincludes applying, during encoding, a forward secondary transform to anoutput of a forward primary transform applied to a residual of a videoblock prior to quantization, or wherein the secondary transform toolincludes applying, during decoding, an inverse secondary transform to anoutput of dequantization to the video block before applying an inverseprimary transform.

13. The method of clause 12, wherein the reduced size corresponds to ½,¼, or ⅛ of the original size of the transform matrix and/or the inversetransform matrix.

14. The method of clause 12, wherein a zero-out region with coefficientszeroed out in the current video block is based on the reduced size ofthe transform matrix and/or the inverse transform matrix.

15. A method for video processing (e.g., method 2740 shown in FIG. 27D),comprising determining (2742), for a conversion between a current videoblock of a video and a coded representation of the video, based on arule, to use a residual pattern among one or more residual patterns,wherein each of the one or more residual patterns corresponds to a maskthat provides information about positions of zeroed out samples; andperforming (2744) a conversion between the video and a codedrepresentation of the video based on the determining.

16. The method of clause 15, wherein each of the one or more residualpatterns includes a portion associated with non-zero transformcoefficients and another portion associated with zero coefficients.

17. The method of clause 15, wherein the rule determines to use theresidual pattern for the current video block having a size of M×Nregardless of a signaled transform index, wherein M and N are equal toor greater than 8.

18. The method of clause 15, wherein the rule specifies whether and/orhow to apply the residual pattern based on a type of a transform toolused in the current video block.

19. The method of clause 18, wherein the determining determines to usethe residual pattern due to a second transform tool used in the currentvideo block, and wherein the secondary transform tool includes applying,during encoding, a forward secondary transform to an output of a forwardprimary transform applied to a residual of a video block prior toquantization, or wherein the secondary transform tool includes applying,during decoding, an inverse secondary transform to an output ofdequantization to the video block before applying an inverse primarytransform.

20. The method of clause 15, wherein the one or more residual patternsare derived from a larger residual pattern with more coefficients.

21. The method of clause 20, wherein, for each of the one or moreresidual patterns, multiple transform matrices are defined.

22. The method of clause 20, wherein an indication of the residualpattern and a transform matrix associated with residual pattern issignaled.

23. The method of clause 20, wherein the one or more residual patternsare derived by excluding positions that are associated with relativelysmall transform coefficients in a K×L transform matrix, and wherein Kand L are integers.

24. The method of clause 23, wherein the one or more residual patternsinclude a first residual pattern that is derived by excluding positionsassociated with first K0 smallest transform coefficients and wherein(K*L−K0) positions are assigned with residuals, wherein K0 is aninteger.

25. The method of clause 24, wherein the one or more residual patternsinclude a second residual pattern that is derived by excluding positionsassociated with first K1 smallest transform coefficients and wherein(K*L−K1) positions are assigned with residuals, wherein K1 is an integerthat is different from K0.

26. The method of clause 15, wherein the current video block has a sizethat is smaller than a maximum transform block.

27. The method of clause 15, wherein the rule specifies to use theresidual pattern based on at least one of a color component or codedinformation of the current video block and/or a neighboring block of thecurrent video block.

28. A method of video processing (e.g., method 2750 shown in FIG. 27E),comprising determining (2752), based on a type of a primary transform, asecondary transform tool associated with a current video block of avideo; and performing (2754) a conversion between the video and a codedrepresentation of the video, wherein the secondary transform toolincludes applying during encoding, a forward secondary transform to anoutput of a forward primary transform applied to a residual of a videoblock prior to quantization, or wherein the secondary transform toolincludes applying, during decoding, an inverse secondary transform to anoutput of dequantization to the video block before applying an inverseprimary transform.

29. The method of clause 28, wherein the determining includes selecting,for the current video block of a video, a transform set or transformmatrices to be used in an application of the secondary transform tool tothe current video block based on a rule, wherein the rule specifies toselect the transform set or the transform matrices based on i) a type ofthe forward primary transform or a type of the inverse primary transformof the current video block, ii) a size of the current video block, oriii) a shape of the current video block.

30. The method of clause 28 or 29, wherein the forward primary transformor the inverse primary transform includes variances of DCT-II, DST-VII,DCT-VIII, transform skip mode, identity transform, or transform fromtraining.

31. The method of clause 29, wherein the rule specifies that a set oftransform matrices is used in a transform skip mode in which a transformis bypassed or an identify transform is applied.

32. The method of clause 29, wherein the coded representation includes asyntax element to indicate whether the second transform tool is used ornot for a transform skip mode or which matrix among the set is used forthe current video block.

33. The method of clause 29, wherein the rule specifies that a set oftransform matrices is used for DCT-II or a first forward or inverseprimary transform and another set of transform matrices is used foranother forward or inverse primary transform.

34. The method of clause 29, wherein the rule specifies that a set oftransform matrices corresponds to a corresponding primary transform modeused in the forward primary transform or the inverse primary transform.

35. The method of clause 29, wherein the rule specifies that a set oftransform matrices used for a primary transform mode includes at leastone matrix that is different from any one in two or more sets oftransform matrices used for different primary transform modes.

36. The method of clause 29, wherein a set of transform matrices usedfor the current video block corresponds to a category that is determinedbased on the size or the shape of the current video block, the categorybeing different from a category of another video block having adifferent size or a different shape from that of the current videoblock.

37. A method of video processing (e.g., method 2760 shown in FIG. 27F),comprising determining (2762) to use multiple sets of transform matricesto be used to process a video region of a video; and performing (2764) aconversion between the video and a coded representation of the video,and wherein the coded representation includes an indication of which setamong the multiple sets is used for the video region.

38. The method of clause 37, wherein the video region corresponds to asequence, a picture, a slice, a tile, a brick, a subpicture, or thevideo.

39. A method for video processing (e.g., method 2770 shown in FIG. 27G),comprising determining (2772), based on a rule, a transform matrix or aninverse transform matrix to be used in an application of a secondarytransform tool to a current video block of a video; and performing(2774) a conversion between the video and a coded representation of thevideo, wherein the secondary transform tool includes applying, duringencoding, a forward secondary transform to an output of a forwardprimary transform applied to a residual of a video block prior toquantization, or wherein the secondary transform tool includes applying,during decoding, an inverse secondary transform to an output ofdequantization to the video block before applying an inverse primarytransform, wherein the rule specifies to determine coefficients of thetransform matrix or the inverse transform matrix based on a variablespecifying a transform output length.

40. The method of clause 39, wherein the coefficients of the transformmatrix or the inverse transform matrix have lower frequencies as thoseof a transform matrix or an inverse transform matrix that is not basedon the variable.

41. The method of clause 39 or 40, wherein the variable is 16 or 48.

42. A method for video processing (e.g., method 2780 shown in FIG. 27H),comprising determining (2782) a transform matrix or an inverse transformmatrix to be used in an application of a secondary transform tool to acurrent video block of a video based on a rule; and performing (2784) aconversion between the video and a coded representation of the video,wherein the secondary transform tool includes applying, during encoding,a forward secondary transform to an output of a forward primarytransform applied to a residual of a video block prior to quantization,or wherein the secondary transform tool includes applying, duringdecoding, an inverse secondary transform to an output of dequantizationto the video block before applying an inverse primary transform, whereinthe rule specifies to determine the transform matrix or the inversetransform matrix by using a retraining process.

43. The method of clause 42, wherein the retraining process includesretraining a reconstructed transform coefficient block and applying aneigen decomposition to calculate eigenvalues and eigenvectors.

44. The method of clause 42, wherein the rule specifies that thetransform matrix or the inverse transform matrix is determined onlyafter encoding or decoding an intra slice or an intra picture of thevideo.

45. The method of clause 42, wherein the transform matrix or the inversetransform matrix is used to replace at least a part of an originaltransform matrix or an original inverse transform matrix that isobtained without using the retraining process.

46. The method of clause 42, wherein the transform matrix or the inversetransform matrix is used as an additional set of original transformmatrices that are obtained without using the retraining process.

47. The method of clause 42, wherein the coded representation includesinformation that corresponds to i) at least a part of a retrainedtransform matrix, ii) first difference values between a retrainedtransform matrix and the transform matrix, or iii) second differencevalues between a retrained transform matrix and a previously signaledtransform matrix.

48. The method of clause 47, wherein the information is included in aslice header, a picture header, or a picture parameter set (PPS), or anadaption parameter set (APS).

49. The method of any of clauses 1 to 48, wherein the secondarytransform tool corresponds to a low frequency non-separable transform(LFNST) tool, and wherein, using the secondary transform tool: duringencoding, a forward secondary transform is applied to an output of aforward primary transform applied to a residual of the chroma blockprior to quantization, or during decoding, an inverse secondarytransform is applied to an output of dequantization of the chroma blockbefore applying an inverse primary transform.

50. The method of any of clauses 1 to 49, wherein the performing of theconversion includes generating the coded representation from the currentvideo block.

51. The method of any of clauses 1 to 49, wherein the performing of theconversion includes generating the current video block from the codedrepresentation.

52. An apparatus in a video system comprising a processor and anon-transitory memory with instructions thereon, wherein theinstructions upon execution by the processor, cause the processor toimplement the method in any one of clauses 1 to 51.

53. A computer program product stored on a non-transitory computerreadable media, the computer program product including program code forcarrying out the method in any one of clauses 1 to 51.

From the foregoing, it will be appreciated that specific embodiments ofthe presently disclosed technology have been described herein forpurposes of illustration, but that various modifications may be madewithout deviating from the scope of the invention. Accordingly, thepresently disclosed technology is not limited except as by the appendedclaims.

Implementations of the subject matter and the functional operationsdescribed in this patent document can be implemented in various systems,digital electronic circuitry, or in computer software, firmware, orhardware, including the structures disclosed in this specification andtheir structural equivalents, or in combinations of one or more of them.Implementations of the subject matter described in this specificationcan be implemented as one or more computer program products, i.e., oneor more modules of computer program instructions encoded on a tangibleand non-transitory computer readable medium for execution by, or tocontrol the operation of, data processing apparatus. The computerreadable medium can be a machine-readable storage device, amachine-readable storage substrate, a memory device, a composition ofmatter effecting a machine-readable propagated signal, or a combinationof one or more of them. The term “data processing unit” or “dataprocessing apparatus” encompasses all apparatus, devices, and machinesfor processing data, including by way of example a programmableprocessor, a computer, or multiple processors or computers. Theapparatus can include, in addition to hardware, code that creates anexecution environment for the computer program in question, e.g., codethat constitutes processor firmware, a protocol stack, a databasemanagement system, an operating system, or a combination of one or moreof them.

A computer program (also known as a program, software, softwareapplication, script, or code) can be written in any form of programminglanguage, including compiled or interpreted languages, and it can bedeployed in any form, including as a stand-alone program or as a module,component, subroutine, or other unit suitable for use in a computingenvironment. A computer program does not necessarily correspond to afile in a file system. A program can be stored in a portion of a filethat holds other programs or data (e.g., one or more scripts stored in amarkup language document), in a single file dedicated to the program inquestion, or in multiple coordinated files (e.g., files that store oneor more modules, sub programs, or portions of code). A computer programcan be deployed to be executed on one computer or on multiple computersthat are located at one site or distributed across multiple sites andinterconnected by a communication network.

The processes and logic flows described in this specification can beperformed by one or more programmable processors executing one or morecomputer programs to perform functions by operating on input data andgenerating output. The processes and logic flows can also be performedby, and apparatus can also be implemented as, special purpose logiccircuitry, e.g., an FPGA (field programmable gate array) or an ASIC(application specific integrated circuit).

Processors suitable for the execution of a computer program include, byway of example, both general and special purpose microprocessors, andany one or more processors of any kind of digital computer. Generally, aprocessor will receive instructions and data from a read only memory ora random access memory or both. The essential elements of a computer area processor for performing instructions and one or more memory devicesfor storing instructions and data. Generally, a computer will alsoinclude, or be operatively coupled to receive data from or transfer datato, or both, one or more mass storage devices for storing data, e.g.,magnetic, magneto optical disks, or optical disks. However, a computerneed not have such devices. Computer readable media suitable for storingcomputer program instructions and data include all forms of nonvolatilememory, media and memory devices, including by way of examplesemiconductor memory devices, e.g., EPROM, EEPROM, and flash memorydevices. The processor and the memory can be supplemented by, orincorporated in, special purpose logic circuitry.

It is intended that the specification, together with the drawings, beconsidered exemplary only, where exemplary means an example. As usedherein, the use of “or” is intended to include “and/or”, unless thecontext clearly indicates otherwise.

While this patent document contains many specifics, these should not beconstrued as limitations on the scope of any invention or of what may beclaimed, but rather as descriptions of features that may be specific toparticular embodiments of particular inventions. Certain features thatare described in this patent document in the context of separateembodiments can also be implemented in combination in a singleembodiment. Conversely, various features that are described in thecontext of a single embodiment can also be implemented in multipleembodiments separately or in any suitable subcombination. Moreover,although features may be described above as acting in certaincombinations and even initially claimed as such, one or more featuresfrom a claimed combination can in some cases be excised from thecombination, and the claimed combination may be directed to a subcombination or variation of a sub combination.

Similarly, while operations are depicted in the drawings in a particularorder, this should not be understood as requiring that such operationsbe performed in the particular order shown or in sequential order, orthat all illustrated operations be performed, to achieve desirableresults. Moreover, the separation of various system components in theembodiments described in this patent document should not be understoodas requiring such separation in all embodiments.

Only a few implementations and examples are described and otherimplementations, enhancements and variations can be made based on whatis described and illustrated in this patent document.

What is claimed is:
 1. A method for processing video data, comprising:performing a conversion between a current video region of a video and abitstream of the video, wherein the performing of the conversionincludes configuring, only based on a partition type of the currentvideo region, a context model for coding a first bin, the first bin anda second bin included in a bin string corresponding to a first index ofa secondary transform tool applied to the current video region, whereinin case that the partition type is not a single tree type, a variablectxInc which is used to determine the context model is set equal to 1,wherein the first index indicates an applicability of the secondarytransform tool and/or a kernel information of the secondary transformtool, and wherein the secondary transform tool includes applying, duringencoding, a forward secondary transform to an output of a forwardprimary transform applied to a residual of a video block prior toquantization, or wherein the secondary transform tool includes applying,during decoding, an inverse secondary transform to an output ofdequantization to the video block before applying an inverse primarytransform.
 2. The method of claim 1, wherein the partition type is asingle tree type or a dual tree type.
 3. The method of claim 1, whereinin case that the partition type is a single tree type, a variable ctxIncwhich is used to determine the context model is set equal to
 0. 4. Themethod of claim 1, wherein the current video region is a current videoblock, and whether the first index is included in the bitstream is basedon a relationship between at least one of a width (W) and a height (H)of the current video block and an allowed maximum transform size (T). 5.The method of claim 4, wherein a location of a last non-zero coefficientin a residual of the current video block is determined based on at leastone syntax element in the bitstream, and whether or how to include thefirst index in the bitstream is based on a location of the last non-zerocoefficient.
 6. The method of claim 5, wherein the first index is notincluded in the bitstream in a case that the last non-zero coefficientis not located in a region of the current video block to which that thesecondary transform tool is applied.
 7. The method of claim 1, whereinin response to the first index indicating the secondary transform toolbeing enabled, a second index indicating an applicability of the forwardprimary transform or the inverse primary transform and a kernelinformation of the forward primary transform or the inverse primarytransform is not present in the bitstream and inferred to be not appliedto the current video region.
 8. The method of claim 1, wherein thesecondary transform tool corresponds to a low frequency non-separabletransform (LFNST) tool.
 9. The method of claim 1, wherein the conversionincludes encoding the current video region into the bitstream.
 10. Themethod of claim 1, wherein the conversion includes decoding the currentvideo region from the bitstream.
 11. An apparatus for processing videodata comprising a processor and a non-transitory memory withinstructions thereon, wherein the instructions upon execution by theprocessor, cause the processor to: perform a conversion between acurrent video region of a video and a bitstream of the video, whereinthe performing of the conversion includes configuring, only based on apartition type of the current video region, a context model for coding afirst bin, the first bin and a second bin included in a bin stringcorresponding to a first index of a secondary transform tool applied tothe current video region, wherein in case that the partition type is nota single tree type, a variable ctxInc which is used to determine thecontext model is set equal to 1, wherein the first index indicates anapplicability of the secondary transform tool and/or a kernelinformation of the secondary transform tool, and wherein the secondarytransform tool includes applying, during encoding, a forward secondarytransform to an output of a forward primary transform applied to aresidual of a video block prior to quantization, or wherein thesecondary transform tool includes applying, during decoding, an inversesecondary transform to an output of dequantization to the video blockbefore applying an inverse primary transform.
 12. The apparatus of claim11, wherein the partition type is a single tree type or a dual treetype; wherein in case that the partition type is a single tree type, avariable ctxInc which is used to determine the context model is setequal to
 0. 13. The apparatus of claim 11, wherein the current videoregion is a current video block, and whether the first index is includedin the bitstream is based on a relationship between at least one of awidth (W) and a height (H) of the current video block and an allowedmaximum transform size (T); wherein a location of a last non-zerocoefficient in a residual of the current video block is determined basedon at least one syntax element in the bitstream, and whether or how toinclude the first index in the bitstream is based on a location of thelast non-zero coefficient; wherein the first index is not included inthe bitstream in a case that the last non-zero coefficient is notlocated in a region of the current video block to which that thesecondary transform tool is applied; wherein in response to the firstindex indicating the secondary transform tool being enabled, a secondindex indicating an applicability of the forward primary transform orthe inverse primary transform and a kernel information of the forwardprimary transform or the inverse primary transform is not present in thebitstream and inferred to be not applied to the current video region;and wherein the secondary transform tool corresponds to a low frequencynon-separable transform (LFNST) tool.
 14. A non-transitorycomputer-readable storage medium storing instructions that cause aprocessor to: perform a conversion between a current video region of avideo and a bitstream of the video, wherein the performing of theconversion includes configuring, only based on a partition type of thecurrent video region, a context model for coding a first bin, the firstbin and a second bin included in a bin string corresponding to a firstindex of a secondary transform tool applied to the current video region,wherein in case that the partition type is not a single tree type, avariable ctxInc which is used to determine the context model is setequal to 1, wherein the first index indicates an applicability of thesecondary transform tool and/or a kernel information of the secondarytransform tool, and wherein the secondary transform tool includesapplying, during encoding, a forward secondary transform to an output ofa forward primary transform applied to a residual of a video block priorto quantization, or wherein the secondary transform tool includesapplying, during decoding, an inverse secondary transform to an outputof dequantization to the video block before applying an inverse primarytransform.
 15. The non-transitory computer-readable storage medium ofclaim 14, wherein the partition type is a single tree type or a dualtree type; wherein in case that the partition type is a single treetype, a variable ctxInc which is used to determine the context model isset equal to
 0. 16. The non-transitory computer-readable storage mediumof claim 14, wherein the current video region is a current video block,and whether the first index is included in the bitstream is based on arelationship between at least one of a width (W) and a height (H) of thecurrent video block and an allowed maximum transform size (T); wherein alocation of a last non-zero coefficient in a residual of the currentvideo block is determined based on at least one syntax element in thebitstream, and whether or how to include the first index in thebitstream is based on a location of the last non-zero coefficient;wherein the first index is not included in the bitstream in a case thatthe last non-zero coefficient is not located in a region of the currentvideo block to which that the secondary transform tool is applied;wherein in response to the first index indicating the secondarytransform tool being enabled, a second index indicating an applicabilityof the forward primary transform or the inverse primary transform and akernel information of the forward primary transform or the inverseprimary transform is not present in the bitstream and inferred to be notapplied to the current video region; and wherein the secondary transformtool corresponds to a low frequency non-separable transform (LFNST)tool.
 17. A method for storing bitstream of a video, comprising:generating the bitstream of the video for a current video region of thevideo; and storing the bitstream in a non-transitory computer-readablerecording medium, wherein the generating the bitstream includesconfiguring, only based on a partition type of the current video region,a context model for coding a first bin, the first bin and a second binincluded in a bin string corresponding to a first index of a secondarytransform tool applied to the current video region, wherein in case thatthe partition type is not a single tree type, a variable ctxInc which isused to determine the context model is set equal to 1, wherein the firstindex indicates an applicability of the secondary transform tool and/ora kernel information of the secondary transform tool, and wherein thesecondary transform tool includes applying, during encoding, a forwardsecondary transform to an output of a forward primary transform appliedto a residual of a video block prior to quantization, or wherein thesecondary transform tool includes applying, during decoding, an inversesecondary transform to an output of dequantization to the video blockbefore applying an inverse primary transform.
 18. The method of claim17, wherein the partition type is a single tree type or a dual treetype; wherein in case that the partition type is a single tree type, avariable ctxInc which is used to determine the context model is setequal to
 0. 19. The method of claim 17, wherein the current video regionis a current video block, and whether the first index is included in thebitstream is based on a relationship between at least one of a width (W)and a height (H) of the current video block and an allowed maximumtransform size (T); wherein a location of a last non-zero coefficient ina residual of the current video block is determined based on at leastone syntax element in the bitstream, and whether or how to include thefirst index in the bitstream is based on a location of the last non-zerocoefficient; wherein the first index is not included in the bitstream ina case that the last non-zero coefficient is not located in a region ofthe current video block to which that the secondary transform tool isapplied; wherein in response to the first index indicating the secondarytransform tool being enabled, a second index indicating an applicabilityof the forward primary transform or the inverse primary transform and akernel information of the forward primary transform or the inverseprimary transform is not present in the bitstream and inferred to be notapplied to the current video region; and wherein the secondary transformtool corresponds to a low frequency non-separable transform (LFNST)tool.